Archimedes compared the area enclosed by a circle to a right triangle whose base has the length of the circumference of the circle and whose height is equal to the radius of the circle. If the area of ​​the circle is not equal to that of the triangle, then it must be larger or smaller. He then eliminates each of these by contradiction, leaving equality as the only possibility. Say no to plagiarism. Get a tailor-made essay on "Why Violent Video Games Shouldn't Be Banned"? Get an original essay Archimedes' test consists of constructing a circle ABCD and a triangle K. Archimedes begins by inscribing a square in the circle and bisects the segments of the arc AB, BC, CD, DE subtended by the sides of the square. Subsequently we proceed to inscribe another polygon on the bisected points. It repeats this process until the difference in the area between the circle and the inscribed polygon is less than the difference between the area of ​​the circle and the area of ​​the triangle. The polygon is therefore greater than the triangle K. Archimedes then proceeds to explain that a line from the center of the polygon to the bisection of one of its sides is shorter than the radius of the circle and its circumference is less than the circumference of the circle. This disproves the claim that the polygon is larger than the triangle, since the legs of the triangle are made up of the radius and circumference of the circle. The triangle K cannot be simultaneously smaller and larger than the polygon, and therefore it cannot be smaller than the circle. Archimedes, after having demonstrated that the triangle cannot be smaller than the circle, continues to demonstrate that the triangle also cannot be larger than the circle. This is achieved by first assuming that the triangle K is larger than the circle ABCD. Then, a square is circumscribed around the circle so that lines drawn from the center of the circle pass through points A, B, C, and D and bisect the corners of the square, one of which Archimedes labels T. Archimedes then connects the sides of the square with a tangent line and labels the points where the line meets the squares G and F. He goes on to say that since TG > GA > GH, the triangle formed by FTG is larger than half the area of ​​the area difference between the square and the circle. Archimedes uses the fact that continued bisecting the arc of a circle will produce a polygon with this characteristic to claim that continuing this method will eventually produce a polygon around the circle such that the difference in area between the polygon and the circle is less than the difference of area between the triangle K and the circle. The polygon therefore has an area smaller than that of the triangle K. The length of a line from the center of the circle to one side of the polygon is equal to the radius of the circle. However, the perimeter of the polygon is greater in length than the circumference of the circle, and since the circumference of the circle is equal to the length of the longest leg of the triangle, the area of ​​the polygon must be greater than the triangle K. Once again, the triangle it cannot be simultaneously larger and smaller than the polygon, therefore the triangle cannot be larger than the circle. Please note: this is just an example. Get a custom paper from our expert writers now. Get a custom essay on Archimedes to prove his theory using contradiction. After proving that the triangle with legs equal to the radius and circumference of a given circle has no area greater or less than that circle, he concludes that the two must be equal in area.