Area of a circle calculator

The area of a circle is A = πr², pi times the radius squared. The radius is the distance from the center to the edge. For a radius of 5: 5² is 25, so the area is 25π, about 78.5398 square units. If the radius is in centimeters, the area is in square centimeters.

The formula has a picture behind it. Cut the circle into many thin wedges, like a pizza, and lay them in a row alternately point-up and point-down. Together they make a shape that looks more and more like a rectangle the thinner the wedges get. Its height is the radius, r, and its length is half the circumference, because half the crust is along the top and half along the bottom: half of 2πr is πr. A rectangle r high and πr long has area πr². For this circle that is 5 by 5π, which is 25π.

Most wrong answers come from the diameter. A circle is usually measured across, and the formula wants half of that. Put the diameter of 10 in where the radius goes and the answer is 100π, about 314.16. That is four times too big, because the radius is squared. If you have the diameter, either halve it first or use A = πd²/4: 100π / 4 = 25π. From the circumference instead, A = C² / (4π); a circumference of 10π gives 100π² / (4π) = 25π, the same circle.

The second slip is squaring the wrong thing. πr² means π × r × r, not (πr)², which would be π² × 25, about 246.74. And area is in square units: 78.54 square centimeters, not centimeters. A circle's area and circumference are different kinds of quantity, and at a radius of 2 they happen to be the same number, 4π, only because of the units.

π is irrational, so the exact answer is 25π and every decimal is a rounding. Taking π as 3.14 gives 78.5, a little low; taking it as 22/7 gives about 78.571, a little high. For a garden bed either is fine. For an exam that says "in terms of π", the answer is 25π, and the table under the result shows each approximation side by side.

It helps to compare a circle with the square around it. A circle of radius 5 fits in a 10-by-10 square of area 100, and fills π/4 of it, about 78.54%. That ratio is the same for every circle, and it is the quick check that an answer is in the right range: a little over three quarters of the diameter squared.

Neighboring formulas build on this one. A ring or washer is the big circle minus the hole, π(R² - r²). A sector, a slice with a given angle, is that fraction of πr², and this calculator's arc mode works it out. An ellipse is a stretched circle with area πab. And because area grows with the square of the radius, a 16-inch pizza has 64π square inches against a 12-inch pizza's 36π: about 78% more.