![]() ![]() Prior to Archimedes, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates, but did not identify the constant of proportionality. The circumference is 2 π r, and the area of a triangle is half the base times the height, yielding the area π r 2 for the disk. Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius in his book Measurement of a Circle. ![]() had found that the area of a disk is proportional to its radius squared. Eudoxus of Cnidus in the fifth century B.C. And the breadth will be the size of the slice. However, the area of a disk was studied by the Ancient Greeks. Hence the length of the rectangle formed will be half of the circumference of the circle which will be r. ![]() Modern mathematics can obtain the area using the methods of integral calculus or its more sophisticated offspring, real analysis. The area of a semicircle is defined as the space enclosed inside a half circle. After substituting the value of d 8, we get, Area of semicircle ( × 8 2 )/8 (3.14 × 8 2 )/8 25.12 cm 2. Therefore, the area of a disk is the more precise phrase for the area enclosed by a circle. The area of a semicircle can be calculated using the formula, Area of semicircle d 2 /8, where 'd' is the diameter. The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = 1 / 2 × 2π r × r, holds for a circle.Īlthough often referred to as the area of a circle in informal contexts, strictly speaking the term disk refers to the interior region of the circle, while circle is reserved for the boundary only, which is a curve and covers no area itself. One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons with an increasing number of sides. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159. In geometry, the area enclosed by a circle of radius r is π r 2. ![]()
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