Area of a Circle Formula

The area of a circle is the amount of flat space it encloses, and it grows with the square of the radius — doubling the radius multiplies the area by four, not two. That's why a 16-inch pizza has roughly four times the food of an 8-inch pizza despite being only twice as wide. The formula A = πr² was known to Greek mathematicians in the 3rd century BCE; Archimedes proved it by bounding the circle between inscribed and circumscribed polygons and showing that as the polygon grew to infinitely many sides its area squeezed onto the circle's.

If you only know the diameter d instead of the radius, remember that r = d/2, so A = π(d/2)² = πd²/4. Both forms give the same answer — use whichever matches the quantity the question actually gives you, rather than converting unnecessarily.

The value of π is irrational (its decimal expansion never terminates or repeats), so an "exact" answer for a circle's area is usually left with π still in it — for example, 25π cm² — and only converted to a decimal if the question asks. Use π ≈ 3.14 for quick estimates, 22/7 for fraction-friendly problems where the radius is a multiple of 7, or π ≈ 3.14159 when you need three or more decimal places of accuracy.

Circle-area questions appear everywhere in exams: finding the cross-section of a pipe in physics problems, computing the grazing area of a tethered animal in aptitude questions, calculating how much paint covers a round table in word problems, or working out the area of an annulus (a ring shape formed by a smaller circle cut out of a larger one — the area is simply the difference of the two πr² values).