How to calculate the area of a circle, plus a simple proof for the formula

The formula for the area of a circle is A = πr², or pi times r squared (radius squared). We use this formula to find the area of circles when the radius is given, or when the diameter is given. We also solve two related word problems, one that involves a circle inside a rectangle, and another that has two concentric circles.

The second video shows a simple proof for the area of a circle. This is an informal derivation for the A = πr² formula for the area of a circle. We start out with a circle divided into 16 equal sectors. The area of each sector is approximated by the area of a triangle. When we calculate the area of 16 such triangles, approximating the base and the altitude of each triangle with 1/16 part of the circumference and the radius, respectively, we arrive at the usual formula for the area of the circle.

What is missing from this informal "proof" is an actual proof of the fact that as you increase the number of sectors, the total area of the approximating triangles does approach the area of the circle. (Such proofs are done in calculus courses, using limits.)

See also

Pi and the circumference of a circle — video lesson

Math Mammoth Geometry 3 — a self-teaching worktext with explanations & exercises (grade 7)

Back to the geometry videos index

Back to 7th grade/pre-algebra videos index

Back to all videos index