Welding essays

Welding essays The development of today's technologies in the industrial world and welding have/_. improved tremendously compared to one hundred years ago. Cars are becoming lighter and airplanes sleeker. The automobile industry has waited for this to come around for years; it is called aluminum welding. This becomes possible with the help of new methods of aluminum welding in many different fields of the auto industry. With more and more technology evolving, aluminum welding is progressing and becoming increasingly important in today's industrial world. Welding within itself has existed for a long time. It dates back to the Egyptians in the middle ages where gold boxes were found that had been welded together. This all evolved out of blacksmithing and later became more sophisticated. Nothing really came around until the late 1800's when arc welding was being experimented with and later invented. When steel became popular with the industrial world, welding also became popular. There was not an efficient way to bring together steel other than just bolting it together (Peter). People knew that those bolts could not possibly last forever. As time progressed, new materials began to be used in the industries and therefore new methods of welding became discovered. The main method found was Arc welding and it just branched off from there. But in aluminum welding, the Friction-stir and Magnetic pulse processes are the two important ones to look at. The breakthrough occurred when aluminum began to be used with airplanes and automobiles. Thus, aluminum welding and these methods started their initial use. With the car industry booming the way it is and new cars coming out almost every half of a year, technology has to keep up with reality. In today's modern world, a lot of aluminum goes into out cars. This is done for durability and the weight of the cars. Because the cars are made of ...

A Look at the Valley and Ridge

A Look at the Valley and Ridge Viewed from above, the Valley and Ridge physiographic province is one of the most defining features of the Appalachian Mountains; its alternating, narrow ridges and valleys almost resemble a corduroy pattern. The province is situated west of the Blue Ridge Mountain province and east of the Appalachian Plateau. Like the rest of the Appalachian Highlands Region, the Valley and Ridge moves from southwest to northeast (from Alabama to New York).   The Great Valley, which makes up the eastern portion of the Valley and Ridge, is known by more than 10 different regional names over its 1,200-mile path. It has hosted settlements on its fertile soils and served as a north-south travel route for a very long time.  The western half of the Valley and Ridge is comprised of the Cumberland Mountains to the south and Allegheny Mountains to the north; the boundary between the two is located in West Virginia. Many mountain ridges in the province rise upwards of 4,000 feet. Geologic Background Geologically, the Valley and Ridge is very different than the Blue Ridge Mountain province, even though the neighboring provinces were shaped during many of the same mountain building episodes and both rise to above-average elevations. The Valley and Ridge rocks are almost entirely sedimentary and were initially deposited during the Paleozoic era. During this time, an ocean covered much of eastern North America.  You can find many marine fossils in the province as evidence, including brachiopods, crinoids and trilobites. This ocean, along with the erosion of bordering landmasses, generated large amounts of sedimentary rock.   The ocean eventually came to a close in the Alleghanian orogeny, as the North American and African protocontinents came together to form Pangea. As the continents collided, the sediment and rock stuck between them had nowhere to go. It was put under stress from the approaching landmass and folded into great anticlines and synclines. These layers were then thrust up to 200 miles westward.   Since mountain building ceased around 200 million years ago, the rocks have eroded to form the present-day landscape. Harder, more erosion-resistant sedimentary rocks like sandstone and conglomerate cap the tops of ridges, while softer rocks like limestone, dolomite and shale have eroded into valleys.  The folds decrease in deformation moving west until they die out underneath the Appalachian Plateau.   Places to See Natural Chimney Park, Virginia - These towering rock structures, reaching heights of 120 feet, are the result of karst topography. Hard columns of limestone rock were deposited during the Cambrian and withstood the test of time as the surrounding rock eroded away.   Folds and faults of Georgia - Dramatic anticlines and synclines can be seen within roadcuts throughout the entire Valley and Ridge, and Georgia is no exception. Check out Taylor Ridge, Rockmart slate folds and the Rising Fawn thrust fault.   Spruce Knob, West Virginia - At 4,863 feet, Spruce Knob is the highest point in West Virginia, the Allegheny Mountains and the entire Valley and Ridge province.   Cumberland Gap, Virginia, Tennessee and Kentucky - Often referenced in folk and blues music, the Cumberland Gap is a natural pass through the Cumberland Mountains. Daniel Boone first marked this trail in 1775, and it served as the gateway to the West into the 20th century.   Horseshoe Curve, Pennsylvania - Although more of a historical or cultural landmark, Horseshoe Curve is a great example of geologys influence on civilization and transportation. The imposing Allegheny Mountains long stood as a barrier to efficient travel across the state. This engineering marvel was completed in 1854 and reduced the Philadelphia-to-Pittsburgh travel time from 4 days to 15 hours.

Dq-4.1-Sheila Coursework Example | Topics and Well Written Essays - 2000 words

Dq-4.1-Sheila - Coursework Example The socio-technical system is one of the new systems that are employed firms to enhance better performance among their teams of workers working at different levels globally. This system coordinates workers easily thus team work is embraced. Other improved communication channels also play a major role here. Such technologies include Blackberry phones, iPhones and android phones. This paper is going to highlight how the communication in departments and teams can be improved by these innovations. A socio-technical system is a combination of technology and the people in the society. It is an operational system that enhances the working together of all the members in the society using the available techniques (Bass, 2012). Due to this reason, most organizations and business firms employ the socio-technical systems that are easy to adopt. They use these systems to make communication an instant thing in their operations despite the distance between workers. There are various types of socio-technical systems that have been put in use by many globally operating firms (Horspool, 2011). Some of them include Skype, online chats, emails and video conferencing. All of these systems require some components to operate effectively. There are some parts that are common to all of the socio-technical systems. These parts are the software, hardware, procedures, and the people using the system (Frederiksen, 2013). In my workplace we use videoconferencing to hold some board meetings with regional managers in different parts of the world. The use of videoconferencing and Skype can be classified as socio-technical. Therefore, the socio-technical system is also in use at my work area. Knowledge workers who are dispersed to far places in the world sometimes face difficulty of communication. The socio-technical systems are some of the few opportunities available for these team members to connect without being disturbed with the distance. Due to this there is need

Patent Law Essay Example | Topics and Well Written Essays - 2000 words

Patent Law - Essay Example Patents are available for most industrially applicable processes and devices. They may cover: Mechanical devices, such as a mousetrap. It also covers methods for doing things, such as the method used for dyeing or bleaching fabrics. It also include chemical compounds, like for example, a new drug and mixtures of compounds, like that of an improved hand cream. Patents can also cover such diverse matters like vaccines for whooping cough, wire-strippers, and chemical processes (Coyle, 2008). The commercial benefit of a granted patent is that it gives the owner the right to prevent others from exploiting, without his consent, the invention for which a patent has been granted (What is n.d.). A granted patent is a property right which can be bought, sold, licensed to others or used as security. The owner of a granted patent might use it to protect a product or service, which he sells. Alternatively, or as well, he may grant a license to one or more parties, usually in exchange for royaltie s (Ibid). Patents in the UK, as elsewhere in the European Economic Area (EEA), have the duration of 20 years from their filing date, subject to payment of renewal fees and not being invalidated. As mentioned above, the duration for the protection of patents in the UK is 20 years and also renewable every 5 years. After this period of 20 year other people are free to produce or copy the invention. The reason why the term is set to 20 years is because the creator should have enough time to reap the rewards of creating his invention, his intellectual property. 20 years is more than enough time to get a market lead on any invention and has been at this length for... This essay describes and presents a study on the topic of patent law. A patent is a government issued right, that is granted to individuals or groups that protects their original inventions from being made, used, or sold by others without their permission for a set period of time. The law that protects and govern the patent in the UK is the Patent Act of 1977. It requires any new inventions to be a new invention; it can’t have existed before the invention was created. The researcher also discusses types of patents that are present in the United Kingdom today and the duration of these patents both in the United Kingdom and in Europe. Patent duration which is "the period in which the patent holder has monopoly rights to their invention, the granting of usage, distribution, and marketing rights to others, and the right to commercial benefit from such for a specific period". A patent is a form of intellectual property that provides the owner with an exclusive right to use and mark et an invention or process. The owner of a patent has the right to prevent others from using the invention or process without permission. For example, Pharmaceutical companies acquire patents in order to protect their drugs from competition. The researcher also analyzes the Patents Act 1977, that implements a statutory regime whereby an employee of a company may become entitled to a measure of financial reward or compensation where the employer has obtained a large benefit from a patented invention made by an employee.

Understanding contesting claims about the pork barrel issue Essay Example for Free

Understanding contesting claims about the pork barrel issue Essay The talk started with the historical background of Priority Development Assistance Fund. It was truly insightful as I have come to learn when and how it was created. It was surprising to know the total amount of the PDAF that legislators get. I don’t think it is very reasonable for them to get that high amount. I stand with those who are pro to the abolition of PDAF since I learned and came to realize that the role of the legislators are to make laws. Why would they need 200 Million to make laws, they definitely use the fund they get for their own benefits. It was also said in the talk that they get kickbacks to every project they would create or start. I dont know how they can live without conscience because of the millions of pesos they steal from the hard work earned money of ordinary Filipino citizens. If the money was properly used we would probably have a better life in our country now. There would be lesser people who experience poverty. If that money was properly used our country would have been more progressive. I am fuming mad for every political leaders who runs for position just to steal millions of money that is supposedly for the betterment of the citizens of our country and our country itself. The money would have been used to address many problems in health, housing, education, agriculture or even national security. It is sad that those type of leaders get elected because of many citizens of our country do not vote wisely or their vote was bought or there was a fraud in the elections. Those political leaders who seek to steal money from the hardworking citizens should be jailed and suffer for the consequence they did. The talk served as an awakening to us students to the reality of politics. The political leaders should do their jobs properly because the ones who suffer from their wrong doings are the ordinary citizens of our country because they don’t get and enjoy the benefits they should have received.

Shattered by Dick Francis :: essays research papers

Gerald Logan and Martin Stuckey met in a jury room and became immediate friends although they share little in common. Martin is a horse jockey who races at the elite English tracks. Logan, who owns and operates Logan Glass, is a gifted glassblower beginning to earn a well-deserved reputation. Even after the trouble began, Logan never blamed Stuckey nor regretted their friendship. On New Yearà ¢Ã¢â€š ¬Ã¢â€ž ¢s Eve, Logan watches Stuckey race at Cheltenham. However, one of the horses Stuckey rides stumbles and falls on top of the jockey, killing him instantly. Before a stunned Logan can leave the track, he receives a videotape from Stuckeyà ¢Ã¢â€š ¬Ã¢â€ž ¢s valet, who says the deceased planned to give it to him after the races. Logan leaves the tape and his storeà ¢Ã¢â€š ¬Ã¢â€ž ¢s receipt on the store counter to go outside and enjoy the new millennium. When he returns, the tape and his money are gone. A couple of days later, thugs confront Logan demanding the tape. They do not believe him when he tells them he no longer possesses the tape. He also knows he is in trouble unless he recovers the tape and gives it to the proper authorities. Analysis   Ã‚  Ã‚  Ã‚  Ã‚  This is the first novel I have read by Dick Francis. When I first picked up the book I thought I was really going to read some garbage because I saw a horse on the front cover. But honestly, the book was average. There are holes in the plot large enough to ride a horse through. For example, would the information that makes the videotape so valuable really be put on a videotape? The primary villains are cartoon-like and there are many instances in which we are asked to believe if glass-blowing is so damn fascinating to all the secondary characters. The plot was negative. I found it to be very unbelievable. It took place on New Yearà ¢Ã¢â€š ¬Ã¢â€ž ¢s Eve, Loganà ¢Ã¢â€š ¬Ã¢â€ž ¢s best friend dies in a race on New Yearà ¢Ã¢â€š ¬Ã¢â€ž ¢s, Loganà ¢Ã¢â€š ¬Ã¢â€ž ¢s friend Lloyd Baxter lies unconscious in Loganà ¢Ã¢â€š ¬Ã¢â€ž ¢s store supposedly from an epilepsy-attack. Shattered by Dick Francis :: essays research papers Gerald Logan and Martin Stuckey met in a jury room and became immediate friends although they share little in common. Martin is a horse jockey who races at the elite English tracks. Logan, who owns and operates Logan Glass, is a gifted glassblower beginning to earn a well-deserved reputation. Even after the trouble began, Logan never blamed Stuckey nor regretted their friendship. On New Yearà ¢Ã¢â€š ¬Ã¢â€ž ¢s Eve, Logan watches Stuckey race at Cheltenham. However, one of the horses Stuckey rides stumbles and falls on top of the jockey, killing him instantly. Before a stunned Logan can leave the track, he receives a videotape from Stuckeyà ¢Ã¢â€š ¬Ã¢â€ž ¢s valet, who says the deceased planned to give it to him after the races. Logan leaves the tape and his storeà ¢Ã¢â€š ¬Ã¢â€ž ¢s receipt on the store counter to go outside and enjoy the new millennium. When he returns, the tape and his money are gone. A couple of days later, thugs confront Logan demanding the tape. They do not believe him when he tells them he no longer possesses the tape. He also knows he is in trouble unless he recovers the tape and gives it to the proper authorities. Analysis   Ã‚  Ã‚  Ã‚  Ã‚  This is the first novel I have read by Dick Francis. When I first picked up the book I thought I was really going to read some garbage because I saw a horse on the front cover. But honestly, the book was average. There are holes in the plot large enough to ride a horse through. For example, would the information that makes the videotape so valuable really be put on a videotape? The primary villains are cartoon-like and there are many instances in which we are asked to believe if glass-blowing is so damn fascinating to all the secondary characters. The plot was negative. I found it to be very unbelievable. It took place on New Yearà ¢Ã¢â€š ¬Ã¢â€ž ¢s Eve, Loganà ¢Ã¢â€š ¬Ã¢â€ž ¢s best friend dies in a race on New Yearà ¢Ã¢â€š ¬Ã¢â€ž ¢s, Loganà ¢Ã¢â€š ¬Ã¢â€ž ¢s friend Lloyd Baxter lies unconscious in Loganà ¢Ã¢â€š ¬Ã¢â€ž ¢s store supposedly from an epilepsy-attack.

The Life of Walter Mitty (Alternate Ending)

The life of Walter Mitty By Gregory Jones 9/17/12 Walter Mitty the Undefeated, inscrutable to the last. â€Å"Walter! † his wife yelled â€Å"what are you doing can’t you see I’m ready to leave! † Ok Mitty replied they got into the car and Mitty turned on the radio. The Football game had just ended, and the announcer was describing the atmosphere in the game it was the 4th quarter with only 30 seconds left on the clock. â€Å"Ok Mitty you’re going to throw to the drag route understand? â€Å"Yeah coach I know what to do I’ve done this before† Mitty ran onto the field into the huddle. â€Å"Ok guys we got 30 seconds left were going for it now, everyone just run down for a Hail Mary. The players stared at him in confusion they all knew the coach wouldn’t have called the play but they had faith in their quarterback to win the games. Mitty slowly walked up to the center and got set, looking at the linebackers and safety. Hike† Mitty dropped back, but the defensive tackle broke through the line and chased him out of the pocket. Mitty rolled out of the pocket knowing there wouldn’t be any open receivers. He pumped fake to make the linebackers drop deeper in coverage before he started running. He juked the first linebacker and hurdled the second. The crowd began to scream and yell as they realized Mitty had got the first down with 20 seconds remaining on the clock. But Mitty didn’t run out of bounds he was going for the win now.The corners and safety know having knowledge that Mitty was running immediately joined the chase. He stiff armed one corner to the ground still in full stride. The crowd was going completely berserk. Mitty had one last man until he scored the winning touchdown. Mitty was on the 5 the safety on the 1 Mitty leaped off the ground reaching the ball out. â€Å"Walter you passed our house! † â€Å" I was going to score† Mitty whispered â€Å"what?!?!? You mus t remember to take your medicine you always daze off when you don’t. †

Second Foundation 17. War

The mayor of the Foundation brushed futilely at the picket fence of hair that rimmed his skull. He sighed. â€Å"The years that we have wasted; the chances we have thrown away. I make no recriminations, Dr. Darell, but we deserve defeat.† Darell said, quietly, â€Å"I see no reason for lack of confidence in events, sir.† â€Å"Lack of confidence! Lack of confidence! By the Galaxy, Dr. Darell, on what would you base any other attitude? Come here-â€Å" He half-led half-forced Darell toward the limpid ovoid cradled gracefully on its tiny force-field support. At a touch of the mayor's hand, it glowed within – an accurate three-dimensional model of the Galactic double-spiral. â€Å"In yellow,† said the mayor, excitedly, â€Å"we have that region of Space under Foundation control; in red, that under Kalgan.† What Darell saw was a crimson sphere resting within a stretching yellow fist that surrounded it on all sides but that toward the center of the Galaxy. â€Å"Galactography,† said the mayor, â€Å"is our greatest enemy. Our admirals make no secret of our almost hopeless, strategic position. Observe. The enemy has inner lines of communication. He is concentrated; can meet us on all sides with equal ease. He can defend himself with minimum force. â€Å"We are expanded. The average distance between inhabited systems within the Foundation is nearly three times that within Kalgan. To go from Santanni to Locris, for instance, is a voyage of twenty-five hundred parsecs for us, but only eight hundred parsecs for them, if we remain within our respective territories-â€Å" Darell said, â€Å"I understand all that, sir.† â€Å"And you do not understand that it may mean defeat.† â€Å"There is more than distance to war. I say we cannot lose. It is quite impossible.† â€Å"And why do you say that?† â€Å"Because of my own interpretation of the Seldon Plan.† â€Å"Oh,† the mayor's lips twisted, and the hands behind his back flapped one within the other, â€Å"then you rely, too, on the mystical help of the Second Foundation.† â€Å"No. Merely on the help of inevitability – and of courage and persistence.† And yet behind his easy confidence, he wondered- What if- Well- What if Anthor were right, and Kalgan were a direct tool of the mental wizards. What if it was their purpose to defeat and destroy the Foundation. No! It made no sense! And yet- He smiled bitterly. Always the same. Always that peering and peering through the opaque granite which, to the enemy, was so transparent. Nor were the galactographic verities of the situation lost upon Stettin. *** The Lord of Kalgan stood before a twin of the Galactic model which the mayor and Darell had inspected. Except that where the mayor frowned, Stettin smiled. His admiral's uniform glistered imposingly upon his massive figure. The crimson sash of the Order of the Mule awarded him by the former First Citizen whom six months later he had replaced somewhat forcefully, spanned his chest diagonally from right shoulder to waist. The Silver Star with Double Comets and Swords sparkled brilliantly upon his left shoulder. He addressed the six men of his general staff whose uniforms were only less grandiloquent than his own, and his First Minister as well, thin and gray – a darkling cobweb, lost in the brightness. Stettin said, â€Å"I think the decisions are clear. We can afford to wait. To them, every day of delay will be another blow at their morale. If they attempt to defend all portions of their realm, they will be spread thin and we can strike through in two simultaneous thrusts here and here.† He indicated the directions on the Galactic model – two lances of pure white shooting through the yellow fist from the red ball it inclosed, cutting Terminus off on either side in a tight arc. â€Å"In such a manner, we cut their fleet into three parts which can be defeated in detail. If they concentrate, they give up two-thirds of their dominions voluntarily and will probably risk rebellion.† The First Minister's thin voice alone seeped through the hush that followed. â€Å"In six months,† he said, â€Å"the Foundation will grow six months stronger. Their resources are greater, as we all know, their navy is numerically stronger; their manpower is virtually inexhaustible. Perhaps a quick thrust would be safer.† His was easily the least influential voice in the room. Lord Stettin smiled and made a flat gesture with his hand. â€Å"The six months – or a year, if necessary – will cost us nothing. The men of the Foundation cannot prepare; they are ideologically incapable of it. It is in their very philosophy to believe that the Second Foundation will save them. But not this time, eh?† The men in the room stirred uneasily. â€Å"You lack confidence, I believe,† said Stettin, frigidly. â€Å"Is it necessary once again to describe the reports of our agents in Foundation territory, or to repeat the findings of Mr. Homir Munn, the Foundation agent now in our†¦ uh†¦ service? Let us adjourn, gentlemen.† Stettin returned to his private chambers with a fixed smile still on his face. He sometimes wondered about this Homir Munn. A queer water-spined fellow who certainly did not bear out his early promise. And yet he crawled with interesting information that carried conviction with it – particularly when Callia was present. His smile broadened. That fat fool had her uses, after all. At least, she got more with her wheedling out of Munn than he could, and with less trouble. Why not give her to Munn? He frowned. Callia. She and her stupid jealousy. Space! If he still had the Darell girl- Why hadn't he ground her skull to powder for that? He couldn't quite put his finger on the reason. Maybe because she got along with Munn. And he needed Munn. It was Munn, for instance, who had demonstrated that, at least in the belief of the Mule, there was no Second Foundation. His admirals needed that assurance. He would have liked to make the proofs public, but it was better to let the Foundation believe in their nonexistent help. Was it actually Callia who had pointed that out? That's right. She had said- Oh, nonsense! She couldn't have said anything. And yet- He shook his head to clear it and passed on.

Apush Era of Jackson Essay

Apush Era of Jackson Essay Apush: Era of Jackson Essay 1828 Election: Jackson defeats John Quincy Adams despite Jackson’s large speculation and scandals. Jackson’s appeal to the poor non-aristocratic citizens gained him his sweeping victory. Martin Van Buren: Jackson’s Secretary of State first term and Vice President Jackson’s second term who also hopes to be Jackson’s successor and has a tense rivalry with Calhoun. John C. Calhoun: Vice President in Jackson’s first term and opposes Van Buren and hopes to win him over in the next election. From South Carolina, and determined to preserve southern interests and ideals such as slavery and anti-industrialism. Peggy Eaton Affair: Jackson dislikes John Eaton’s wife as she married Senator Eaton shortly before his appointment to Secretary of War. Much hard feeling eventually caused her to withdraw from Washington. Maysville Road Bill 1830: The government is authorized to buy stock in road construction in Maysville, Kentucky to Lexington. How ever Jackson vetoes the bill on the grounds that it is unconstitutional. Many see this as an abuse of power. National Road: Jackson supports and shows limited federal support for transportation movements that are not at a national level. The Tariff of 1828: â€Å"The tariff of abominations† largely opposed to the new tariff as it makes it harder for Britain, France, and other European countries to buy southern, especially South Carolina. South Carolina Exposition and Protest: by John Calhoun opposing Jackson’s new tariff of 1828. Nullification: Calhoun’s theory that states can repeal a federal law and declare it unconstitutional. According to Calhoun, the government could decide to void the law or propose an amendment. Tariffs for protection were considered unconstitutional. Webster-Hayne Debate: Hayne argued that the federal government cannot limit state rights, while Daniel Webster claimed that a country cannot by unified unless all laws are equally enfor ced. Tariff of 1832: Jackson calms nullifiers in South Carolina by reducing tariffs on goods with less competition. South Carolina Ordinance: South Carolinians worried that the tariffs could end slavery and repudiated the Acts of 1828 and 1832. Force Bill: Jackson’s use of the army to enforce the state’s agreement to federal law after he attempted asking congress to lower the tariffs. Eventually both sides agreed to significantly reducing the tariffs by 1842. 1830 Indian Removal Act: Indians are given federal land in the west in exchange for their territories mainly in the Midwest and southeast regions. Black Hawk War 1832: War erupted in Wisconsin/Illinois territory when the Illinois militia forcefully and brutally, killing many, pushed the Sauk and Fox northwest. 1835 Seminole War: The Seminoles and Cherokees in the south fought a guerilla war against the federal removal policy. Most eventually moved west. The Trail of Tears: used by 17,000 Cherokees, this 800 -mile trail with cruelty of soldiers and private owners. 4,000 died on the brutal journey and only 8,000 completed the trip to Oklahoma. Cherokee Nation v. Georgia: The supreme court rules that it does not have the jurisdiction to determine who owns the land of northern Georgia containing newly discovered gold deposits. The judge did say that the

Polygons on ACT Math Geometry Formulas and Strategies

Polygons on ACT Math Geometry Formulas and Strategies SAT / ACT Prep Online Guides and Tips Questions about both circles and various types of polygons are some of the most prevalent types of geometry questions on the ACT. Polygons come in many shapes and sizes and you will have to know them inside and out in order to take on the many different types of polygon questions the ACT has to offer. The good news is that, despite their variety, polygons are often less complex than they look; a few simple rules and strategies are all that you need when it comes to solving an ACT polygon question. This will be your complete guide to ACT polygons- the rules and formulas for various polygons, the kinds of questions you’ll be asked about them, and the best approach for solving these types of questions. What is a Polygon? Before we go to polygon formulas, let’s look at what exactly a polygon is. A polygon is any flat, enclosed shape that is made up of straight lines. To be â€Å"enclosed† means that the lines must all connect, and no side of the polygon can be curved. Polygons NOT Polygons Polygons come in two broad categories- regular and irregular. A regular polygon has all equal sides and all equal angles, while irregular polygons do not. Regular Polygons Irregular Polygons A polygon will always have the same number of sides as it has angles. So a polygon with nine sides will have nine angles. The different types of polygons are named after their number of sides and angles. A triangle is made of three sides and three angles (â€Å"tri† meaning three), a quadrilateral is made of four sides (â€Å"quad† meaning four), a pentagon is made of five sides (â€Å"penta† meaning five), etc. Many of the polygons you’ll see on the ACT (though not all) will either be triangles or some sort of quadrilateral. Triangles in all their forms are covered in our complete guide to ACT triangles, so let’s move on to look at the various types of quadrilaterals you’ll see on the test. Barber shop quartets, quadrilaterals- clearly the secret to success is in fours. Quadrilaterals There are many different types of quadrilaterals, most of which are subcategories of one another. Parallelogram A parallelogram is a quadrilateral in which each set of opposite sides is both parallel and congruent (equal) with one another. The length may be different than the width, but both widths will be equal and both lengths will be equal. Parallelograms are peculiar in that their opposite angles will be equal and their adjacent angles will be supplementary (meaning any two adjacent angles will add up to 180 degrees). Most questions that require you to know this information are quite straightforward. For example: If we draw this parallelogram, we can see that the two angles in question are supplementary. This means that the two angles will add up to 180 degrees. Our final answer is F, add up to 180 degrees. Rhombus A rhombus is a type of parallelogram in which all four sides are equal and the angles can be any measure (so long as their adjacents add up to 180 degrees and their opposite angles are equal). Rectangle A rectangle is a special kind of parallelogram in which each angle is 90 degrees. The rectangle’s length and width can either be equal or different from one another. Square If a rectangle has an equal length and width, it is called a square. This means that a square is a type of rectangle (which in turn is a type of parallelogram), but NOT all rectangles are squares. Trapezoid A trapezoid is a quadrilateral that has only one set of parallel sides. The other two sides are non-parallel. Kite A kite is a quadrilateral that has two pairs of equal sides that meet one another. You'll notice that a lot of polygon definitions will fit inside other definitions, but a little organization (and dedication) will help keep them straight in your head. Polygon Formulas Though there are many different types of polygons, their rules and formulas build off of a few basic ideas. Let’s go through the list. Area Formulas Most polygon questions on the ACT will ask you to find the area or the perimeter of a figure. These will be the most important area formulas for you to remember on the test. Area of a Triangle $$a = {1/2}bh$$ The area of a triangle will always be half the amount of the base times the height. In a right triangle, the height will be equal to one of the legs. In any other type of triangle, you must drop down your own height, perpendicular from the vertex of the triangle to the base. Area of a Square $$l^2$$ Or $$lw$$ Because each side of a square is equal, you can find the area by either multiplying the length times the width or simply by squaring one of the sides. Area of a Rectangle $$lw$$ For any rectangle that is not a square, you must always multiply the base times the height to find the area. Area of a Parallelogram $$bh$$ Finding the area of a parallelogram is exactly the same as finding the area of a rectangle. Because a parallelogram may slant to the side, we say we must use its base and its height (instead of its length and width), but the principle is the same. You can see why the two actions are equal if you were to transform your parallelogram into a rectangle by dropping down straight heights and shifting the base. Area of a Trapezoid $$[(l_1 + l_2)/2]h$$ In order to find the area of a trapezoid, you must find the average of the two parallel bases and multiply this by the height of the trapezoid. Let's take a look at this formula in action, The trapezoid is divided into a rectangle and two triangles. Lengths are given in inches. What is the combined area of the two shaded triangles? A. 4 B. 6 C. 9 D. 12 E. 18 If you remember your formula for trapezoids, then we can find the area of our triangles by finding the area of the trapezoid as a whole and then subtracting out the area of the rectangle inside it. First, we should find the area of the trapezoid. $[(l_1 + l_2)/2]h$ $[(6 + 12)/2]3$ $(18/2)3$ $(9)3$ $27$ Now, we can find the area of the rectangle. $6 * 3$ 18 And finally, we can subtract out the area of the rectangle from the trapezoid. $27 - 18$ 9 The combined area of the triangles is 9. Our final answer is C, 9. In general, the best way to find the area of different kinds of polygons is to transform the polygon into smaller and more manageable shapes. This will also help you if you forget your formulas come test day. For example, if you forget the formula for the area of a trapezoid, turn your trapezoid into a rectangle and two triangles and find the area for each. Luckily for us, this has already been done in this problem. We know that we can find the area of a triangle by ${1/2}bh$ and we already have a height of 3. We also know that the combined bases for the triangles will be: $12 - 6$ 6 So let us say that one triangle has a base of 4 and the other has a base of 2. (Why those numbers? Any numbers for the triangle bases will work so long as they add up to 6.) Now, let us find the area for each triangle. or the first triangle, we have: ${1/2}(4)(3)$ $(2)(3)$ $6$ And for the second triangle, we have: ${1/2}(2)(3)$ $(1)(3)$ 3 Now, let us add them together. $6 + 3$ 9 Again, the area of our triangles together is 9. Our final answer is C, 9. Always remember that there are many different ways to find what you need, so don’t be afraid to use your shortcuts! Side and Angle Formulas Whether your polygon is regular or irregular, the sum of its interior degrees will always follow the rules of that particular polygon. Every polygon has a different degree sum, but this sum will be consistent, no matter how irregular the polygon. For example, the interior angles of a triangle will always equal 180 degrees, whether the triangle is equilateral (a regular polygon), isosceles, acute, or obtuse. So by that same notion, the interior angles of a quadrilateral- whether kite, square, trapezoid, or other- will always add up to be 360 degrees. Interior Angle Sum You will always be able to find the sum of a polygon’s interior angles in one of two ways- by memorizing the interior angle formula, or by dividing your polygon into a series of triangles. Method 1: Interior Angle Formula $$(n−2)180$$ If you have an n number of sides in your polygon, you can always find the interior degree sum by the formula $(n - 2)$ times 180 degrees. Method 2: Dividing Your Polygon Into Triangles The reason the above formula works is because you are essentially dividing your polygon into a series of triangles. Because a triangle is always 180 degrees, you can multiply the number of triangles by 180 to find the interior degree sum of your polygon, whether your polygon is regular or irregular. As we saw, we have two options to find our interior angle sum. Let us try each method. Solving Method 1: formulas $(n - 2)180$ There are 5 sides, so if we plug that into our formula for $n$, we get: $(5 - 2)180$ $3(180)$ 540 Now we can find the sum of the rest of the angle measurements by subtracting our known degree measure, 50, from our total interior degrees of 540. $540 - 50$ 490 Our final answer is K, 490. Solving Method 2: diving polygon into triangles We can also always divide our polygon into a series of triangles to find the total interior degree measure. We can see that our polygon makes three triangles and we know that a triangle is always 180 degrees. This means that the polygon will have a interior degree sum of: $3 * 180$ 540 degrees. And finally, let us subtract the known angle from the total in order to find the sum of the remaining degrees. $540 - 50$ 490 Again, our final answer is K, 490. Individual Interior Angles If your polygon is regular, you will also be able to find the individual degree measure of each interior angle by dividing the degree sum by the number of angles. (Note: n can be used for both the number of sides and the number of angles because the number of sides and angles in a polygon will always be equal.) ${(n - 2)180}/n$ Again, you can choose to either use the formula or the triangle dividing method by dividing your interior sum by the number of angles. Number of Sides As we saw earlier, a regular polygon will have all equal side lengths. And if your polygon is regular, you can find the number of sides by using the reverse of the formula for finding angle measures. A regular polygon with n sides has equal angles of 140 degrees. How many sides does the figure have? 6 7 8 9 10 For this question, it will be quickest for us to use our answers and work backwards in order to find the number of sides in our polygon. (For more on how to use the plugging in answers technique, check out our guide to plugging in answers). Let us start at the middle with answer choice C. We know from our angle formula (or by making triangles out of our polygons) that an eight sided figure will have: $(n - 2)180$ $(8 - 2)180$ $(6)180$ 1080 degrees. Or again, you can always find your degree sum by making triangles out of your polygon. This way you will still end up with (6)180=1080 degrees. Now, let us find the individual degree measures by dividing that sum by the number of angles. $1080/8$ $135$ Answer choice C was too small. And we also know that the more sides a figure has, the larger each individual angle will be, so we can cross off answer choices A and B, as those answers would be even smaller. (How do we know this? A regular triangle will have three 60 degree angles, a square will have four 90 degree angles, etc.) Now let us try answer choice D. $(n - 2)180$ $(9 - 2)180$ $(7)180$ 1260 Or you could find your internal degree sum by once again making triangles from your polygons. Which would again give you $(7)180 = 1260$ degrees. Now let’s divide the degree sum by the number of sides. $1260/9$ $140$ We have found our answer. The figure has 9 sides. Our final answer is D, 9. Number of Diagonals $${n(n - 3)}/2$$ It is common for the ACT to ask you about the number of distinct diagonals in a polygon. Again, you can find this information using the formula or by drawing it out (or a combination of the two). This is basically the same as dividing your polygon into triangles, but they will be overlapping and you are counting the number of lines drawn instead of the number of triangles. Method 1: formula In order to find the number of distinct diagonals in a polygon, you can simply use the formula ${n(n - 3)}/2$, wherein $n$ is the number of sides of the polygon. Method 2: drawing it out The reason the above formula works is a matter of logic. Let’s look at an octagon, for example. You can see that an octagon has eight angles (because it has eight sides). If you were to draw all the diagonals possible from one particular angle, you could draw five lines. You will always be able to draw n−3 lines because one of the angles is being used to form all the diagonals and the lines to the two adjacent angles make up part of the perimeter of the polygon and are therefore NOT diagonals. So you can only draw diagonals to n−3 corners. Now, let’s mark another angle’s series of diagonals. You can see that none of these diagonals overlap, BUT if we were to draw the diagonals from an opposite corner, we would have multiple overlapping diagonals. The adjacent angles will not overlap, but the opposite ones will. This means that there will only be half as many diagonals as the total number of angles multiplied by their possible diagonals (in other words half of n(n−3). This is why our final formula is: ${n(n - 3)}/2$ This is all the angles multiplied by their total number of diagonals, all divided by half so that we do not get overlapping diagonal lines. (Note: of course an alternative to using any form of the formula is to simply draw out your diagonals, making sure to be very very careful to not create any overlapping diagonal lines.) Just make sure you don't dizzy yourself keeping track of all your angles and diagonals. Typical Polygon Questions Now that we’ve been through all of our polygon rules and formulas, let’s look at a few different types of polygon questions you’ll see on the ACT. About half of ACT polygon questions you’ll see will involve diagrams and about half will be word problems. Most all of the word problems will involve quadrilaterals in some form or another. Typically, you will be asked to find one of three things in a polygon question: The measure of an angle (or the sum of two or more angles) The perimeter of a figure The area of a figure Let’s look at a few real ACT math examples of these different types of questions. 1. Finding the measure of an angle We know that we can find the degree measure of a regular polygon by finding their total number of degrees and dividing that by the number of sides/angles. So let us find the sum of the interior degrees of our pentagon. A pentagon can be divided into three triangles, so we know that it has a total of: 3(180) 540 degrees. If we divide this number by the number of sides/angles in a pentagon, we can see that each angle measure is: $540/5$ 108 Now, we also know that every straight line is 180 degrees. This means that we can find the exterior angles of the pentagon by subtracting the interior angles from 180. $180 - 108$ 72 We also know that a triangle's interior degrees always add up to 180, so we can find our final angle by subtracting our two known angles from 180. $180 - 72 - 72$ 36 Our final answer is C, 36. 2: Finding the perimeter of a figure We know that a square has, by definition, all equal sides. Because DC is 6, that means that ED, EB, and BC are all equal to 6 as well. We also know that an equilateral triangle has all equal sides. Because EB equals 6 and is part of the equilateral triangle, EB, AE, and AB are all equal to 6 as well. And, finally, the perimeter of the figure is made up of lines DE, EA, AB, BC, and CD. This means that our perimeter is: 6 + 6 + 6 + 6 + 6 30 Our final answer is C, 30. 3: Using or finding the area of the figure We know that the area of a rectangle is found by multiplying the length times the width, and we also know that a rectangle has two paris of equal sides. So we need to find measurements for the sides that, in pairs, add up to 24 and, when multiplied, will make a prouct of 32. One way we can do this is to use the strategy of plugging in answers. Let us, as usual when using this strategy, start with answer choice C. So, if we have a short side length of 3, we need to double it to find how much the short sides contribute to the total perimeter. $3 * 2$ 6 If we subtract this from our total perimeter, we find that the sum of our longer sides are: $24 - 6$ 18 Which means that each of the longer sides is: $18/2$ 9 Now, if one side length is 3 and the other is 9, then the area of the rectangle will be: $3 * 9$ 27 This is too small to be our area. We need the shorter side lengths to be longer than 3 so that the product of the length and the width will be larger. Let us try option J instead. If we have two side lengths that each measure 4, they will add a total of: $4 * 2$ 8 Now let us subtract this from the total perimeter. $24 - 8$ 16 This is the sum of the longer side lengths, which means we must divide this number in half to find the individual measures. $16/2$ 8 And finally, let us multiply the length times the width to find the area of the rectangle. $8 * 4$ 32 These measurements fit our requirements, which means that the shorter sides must each measure 4. Our final answer is J, 4. Now let's look at the strategies for success for your polygon questions (as well as what to avoid doing). How to Solve a Polygon Question Now that we’ve seen the typical kinds of questions you’ll be asked on the ACT and gone through the process of finding our answers, we can see that each solving method has a few techniques in common. In order to solve your polygon problems most accurately and efficiently, take note of these strategies: #1: Break up figures into smaller shapes Don’t be afraid to write all over your diagrams. Polygons are complicated figures, so always break them into small pieces when you can. Break them apart into triangles, squares, or rectangles and you’ll be able to solve questions that would be impossible to figure out otherwise. Alternatively, you may need to expand your figures by providing extra lines and creating new shapes in which to break your figure. Just always remember to disregard these false lines when you’re finished with the problem. If we create and expand new lines in our figure, we can make our lengths and sides a little more clear. We can also see why this works because our red lines are essentially extensions of the perimeter branching outwards in order to give us a clearer picture. Now, we know that, because the bottom-most horizontal line is equal to 20, the sum of all the other horizontal lines is also equal to 20. We can also see that all the vertical lines will add up to: 12 + 8 + 8 + 12 This means that our total perimeter will be: 20 + 20 + 12 + 12 + 8 + 8 80 Our final answer is B, 80. #2: Use your shortcuts If you don’t feel comfortable memorizing formulas or if you are worried about getting them wrong on test day, don’t worry about it! Just understand your shortcuts (for example, remember that all polygons can be broken into triangles) and you’ll do just fine. #3: When possible, use PIA or PIN Because polygons involve a lot of data, it can be very easy to confuse your numbers or lose track of the path you need to go down to solve the problem. For this reason, it can often help you to use either the plugging in answer strategy (PIA) or the plugging in numbers strategy (PIN), even though it can sometimes take longer (for more on this, check out our guides to PIA and PIN). #4: Keep your work organized There is a lot of information to keep track of when working with polygons (especially once you break the figure into smaller shapes). It can be all too easy to lose your place or to mix-up your numbers, so be extra vigilant about your organization and don’t let yourself lose a well-earned point due to careless error. Before you go ahead and put your polygon knowledge to the test, take a moment to bask in some much-needed Cuteness. Test Your Knowledge Now, let's test your knowledge on polygons with some real SAT math examples. 1. 2. 3. 4. Answers: D, C, G, G Answer Explanations: 1. In order to find the number of distinct diagonals, we can, as always, either use our diagonal formula or be very (very) careful to draw our own. Let us try both methods. Method 1: formula ${n(n - 3)}/2$ We have a hexagon, so there are 6 sides. We can therefore plug 6 in for n. ${6(6 - 3)}/2$ $6(3)/2$ $18/2$ $9$ There will be 9 distinct diagonals. Our final answer is D, 9. Method 2: drawing it out If we draw our own diagonals, we can see that there are still 9 diagonals total. We can color-code these lines here, but you will not have that option on the test, so make sure you are both able to draw out all your diagonals and not count repeat lines. When done correctly, we will have 9 distinct diagonals in our hexagon. Our final answer is D, 9. 2. We know that, by definition, a parallelogram has two pairs of equal sides. So if one side measures 12, then at least one of the other three sides must also measure 12. So let us first subtract our pair of 12-length sides from our total perimeter of 72. $72 - 12 -12$ 48 The remaining pair of sides will have a sum of 48. We also know that the remaining pair of sides must be equal to one another, so let us divide this sum in half in order to find their individual measures. $48/2$ 24 This means that our parallelogram will have side measures of: 12, 12, 24, 24 Our final answer is C. 3. We are told that each of these rectangles is a square, which means that the side lengths for each square will be equal. We also know that, in order to find the area of a square, we can simply square (multiply a number by itself) one of the sides. So, if the larger square has an area of 50 square centimeters, that means that one of the side lengths squared must be equal to 50. In other words: $s^2 = 50$ $s =√50$ $s =√25 *√2$ $s = 5√2$ (For more info on how to manipulate roots and squares like this, check out our guide to ACT advanced integers.) So now we know that the length of each of the sides of the larger square is $5√2$. We also know that the area of the smaller square is 18 and that the length of one of the sides of the shorter square is the length of the side of the larger square, minus x. img src="http://cdn2.hubspot.net/hubfs/360031/body_square_example.png" alt="body_square_example" style="display: block; margin-left: auto; margin-right: auto; width: 212px;" width="212" So let us find x by using this information. $(5√2 - x)^2 = 18$ $5√2 - x =√18$ $5√2 - x =√9 *√2$ $5√2 - x = 3√2$ $-x = -2√2$ $x = 2√2$ We have successfully found the length of $x$. Our final answer is G,$2√2$. 4. We have a few different ways to solve this problem, but one of the easiest is to use the strategy of plugging in our own numbers. This will help us to visualize the lengths and areas much more solidly. So let us imagine for a minute that the longest length of our rectangle is 12 and the shorter side is 4. (Why those numbers? Why not! When using PIN, we can choose any numbers we want to, so long as they do not contradict our given information. And these numbers do not, which means we're good to go.) Now, to make life even simpler, let us divide our rectangle in half and just work with one half at a time. Now, because we have divided our rectangle exactly in half (and we know that we did this because we are told that F and E are both midpoints of the longest side of our rectangle), we know that BF must be 6. Now we have four triangles, three of which are shaded. In order to find the ratio of unshaded area to shaded area, let us find the areas of each of our triangles. To find the area of a triangle, we know we need: ${1/2}bh$ If we take the triangle on the left, we already know that our base is 4. We also know that the height must be 3. Why? Because point G is directly in the middle of our rectangle, so the height will be exactly half of the line BF. This means that our left-most triangle will have an area of: ${1/2}bh$ ${1/2}(4)(3)$ $(2)(3)$ $6$ Now, we know that our right-most triangle (the unshaded triangle) will ALSO have an area of 6 because its height and base will be exactly the same as our left triangle. So let us find the areas of our top and bottom triangles. Again, we already have a given value for our base (in this case 6) and the height will be exactly half of the line BA. This means that the area of our top triangle (as well as our bottom triangle) will be: ${1/2}bh$ ${1/2}(6)(2)$ $(3)(2)$ $6$ Both the left and the top-most triangles have an area of 6, which means that ALL the triangles have equal areas. There is 1 unshaded triangle and 3 shaded triangles. This means that the ratio of unshaded to shaded triangles is 1:3. We also know that this will be the same ratio if we were to complete the problem for the other half of the rectangle. Why? We cut the shape exactly in half, so the ratio of all the unshaded triangles to shaded triangles will be: 2:6 Or, again: 1:3 Our final answer is G, 1:3. A little practice, a little flare, and you've got the path down to all your right answers. The Take Aways Once you internalize the few basic rules of polygons, you’ll find that these questions are not generally as difficult as they may appear at first blush. You may come across irregular polygons and ones with many sides, but the basic strategies and formulas will always be the same. Remember your strategies, keep your work well organized, and know your key definitions, and you will be able to take on even the most difficult polygon questions the ACT can throw at you. What’s Next? You've mastered polygons and now you're raring to take on more (we're guessing). Luckily for you, there are so many more math topics to cover! Take a glance through all the math topics that will appear on the ACT to make sure you've got them locked down tight. Then go ahead and check out our ACT math guides to brush up on any topics you might be rusty on. Feeling nervous about circle questions? Roots and exponents? Fractions and ratios? Whatever you need, we have the guide for you. Want to learn some of the most useful math strategies on the test? Check out our guides to plugging in answers and plugging in numbers to help you solve questions that may have had you scrambling before. Want to get a perfect score? Look no further than our guide to getting a perfect 36 on ACT math, written by a perfect-ACT-scorer. Want to improve your ACT score by 4 points? Check out our best-in-class online ACT prep program. We guarantee your money back if you don't improve your ACT score by 4 points or more. Our program is entirely online, and it customizes what you study to your strengths and weaknesses. If you liked this Math lesson, you'll love our program. Along with more detailed lessons, you'll get thousands of practice problems organized by individual skills so you learn most effectively. We'll also give you a step-by-step program to follow so you'll never be confused about what to study next. Check out our 5-day free trial:

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