Area of a circle calculator

Type the diameter as d = 12, the radius as r = 6 or the circumference as C = 37.7, and the area is the answer at the top, in square units. A plain number counts as the radius unless you change "A plain number is".

A 12-inch pizza comes out as 36π in², about 113.0973 in². Change Units for cm, m or ft, and Decimal places for more or less rounding.

To find the area of a slice, add the angle, as in d = 12, θ = 45°, and set Answer to Sector area.

A = πr². Square the radius, then multiply by π.

From the diameter, halve it first, or use A = πd²/4. The two agree: a diameter of 12 gives π × 6² = 36π, and π × 144/4 = 36π.

From the circumference, A = C²/4π. That one's handy when you can wrap a tape round something but can't find its middle, like a tree or a pipe. A trunk 2 m round covers 4/4π = 1/π m², about 0.3183 m².

Squaring is what catches people out. Double the radius and the area goes up four times; a 16-inch pizza is 64π in², 78% more than a 12-inch one.

Archimedes proved it with a triangle. In Measurement of a Circle he showed that a circle has the same area as a right triangle whose base is the circumference and whose height is the radius.

That triangle's area is ½ × 2πr × r, which is πr². His proof squeezed the circle between polygons inside and outside it and showed the area could be neither more nor less than the triangle's. Modern mathematicians don't count it as rigorous, since it leans on comparisons of curves that look obvious but weren't proved.

You can see a rough version by cutting a circle into thin slices and laying them alternately point up and point down. They make something close to a rectangle, half the circumference long and one radius tall.