Area of a Circle

A circle may look like a simple shape, but measuring the space inside it involves an important idea. The amount of area inside a circle depends on its radius, but not in a one-to-one way. If the radius becomes larger, the area grows much faster because area measures a two-dimensional region.

Why the Area Must Scale with r²

Suppose every length in a figure is multiplied by the same number k. Its width becomes k times larger and its height also becomes k times larger, so its area becomes k² times as large. A circle follows the same rule. This means the area of a circle must depend on the square of its radius, so we can write A = kr², where k is a constant that is the same for every circle. The next question is to determine the value of this constant.

Historical Approximations

Ancient methods approximated the circle by other shapes or constructed squares of nearly equal area. The chapter notes related ideas in the Baudhāyana Śhulbasūtra and later Greek work. Eventually Archimedes showed that the constant is exactly π.

Archimedes' Connection Between Area and Circumference

Archimedes expressed the circle's area as the area of a right triangle whose base is the circumference and whose height is the radius.

Polygon Thought Experiment

For a regular polygon surrounding an inscribed circle, the polygon can be divided into triangles whose common height is the circle's radius. The total area is therefore one-half of the polygon perimeter times r. As the number of polygon sides increases, its boundary approaches the circle and its perimeter approaches the circumference.

Nīlakaṇṭha's Slice Rearrangement

A particularly visual argument slices the circle into many thin sectors and alternates them. As the slices become thinner, the rearranged figure approaches a parallelogram. Its base approaches half the circumference, πr, and its height is r.