Archimedes compared the area surrounded 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 either be greater or less. He then eliminates each of them by contradiction, leaving only equality as the only possibility. Say no to plagiarism. Get a tailor-made essay on “Why violent video games should not be banned”?Get the original essay Archimedes' proof involves constructing a circle ABCD and a triangle K. Archimedes begins by inscribing a square in the circle and cuts the arc segments in two. AB, BC, CD, DE subtended by the sides of the square. Then, it proceeds to inscribe another polygon on the bisected points. It repeats this process until the difference in 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 then larger than the triangle K. Archimedes then explains that a line going 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 smaller than the circumference of the circle. This disproves the claim that the polygon is larger than the triangle, since the legs of the triangle consist of the radius and circumference of the circle. Triangle K cannot be both smaller and larger than the polygon, and therefore cannot be smaller than the circle. After Archimedes proves that the triangle cannot be smaller than the circle, he goes on to prove that the triangle cannot be larger than the circle either. This is accomplished by first assuming that triangle K is larger than circle ABCD. Next, a square is circumscribed around the circle such 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 marks the points where the line meets the square G and F. He goes on to say that because TG > GA > GH, the triangle formed by FTG is larger than half of the area of ​​the square. difference in area between the square and the circle. Archimedes uses the fact that the continuous bisector of the arc of a circle will produce a polygon with this characteristic to argue that continuing this method will ultimately produce a polygon around the circle such that the difference in area between the polygon and the circle is less than the difference in area between the triangle K and the circle. The polygon therefore has an area less than that of the triangle K. The length of a line going 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 polygon must have a larger area than triangle K. Again, the triangle cannot be both larger and smaller than the polygon, so the triangle cannot be larger than the circle. Keep in mind: this is just a sample. Get a personalized article from our expert writers now. Get a custom essay Archimedes accomplished to prove his theory using contradiction. After having proven that the triangle whose branches are equal to the radius and the circumference of a given circle has neither more nor less area than this circle, he concludes that the two must be of the same area..