The area of a circle: why it is πr², with worked examples

The area of a circle is π times the radius squared: A = πr², where π ≈ 3.14159. A circle of radius 5 cm has an area of π × 25 ≈ 78.54 cm². If you know the diameter instead, halve it first; if you know the circumference, divide by 2π to get the radius. The circle calculator I built works from any one of these measurements, and this guide explains where the formula comes from.

The formula

The radius r is the distance from the centre to the edge, and the diameter d is twice that. The two key formulas are:

• Area: A = πr², or equivalently πd² ÷ 4.
• Circumference: C = 2πr, or equivalently πd.

Area is measured in square units, so a radius in centimetres gives an area in square centimetres. Squaring the radius is the step people most often forget; πr² means π × r × r, not (πr)².

Why the area is πr²

Cut a circle into many thin slices, like a pizza, and lay them side by side, alternately pointing up and down. The result looks roughly like a parallelogram. Its height is the radius r, and its base is half of the circumference, because half the crusts face up and half face down: ½ × 2πr = πr. The area is base times height, πr × r = πr². The more slices you cut, the straighter the edges and the closer the shape is to a true rectangle.

Archimedes gave a rigorous argument in the third century BC. He compared the circle with a right triangle whose base is the circumference and whose height is the radius, with area ½ × 2πr × r = πr², and showed using polygons inside and outside the circle that the circle's area can be neither larger nor smaller than that triangle's.

Circumference and π

π is defined as the ratio of any circle's circumference to its diameter, the same for every circle. It is irrational, so its decimals never end or repeat; 22/7 ≈ 3.1429 and 3.14 are common approximations, good to about 0.04% and 0.05%. The calculator uses π to full double precision. A pleasing consequence of C = 2πr: if you wrapped a rope tightly around the Earth and then added 1 m to its length, it could be raised evenly by 1 ÷ 2π ≈ 0.159 m, about 16 cm, all the way round. The same would be true for a football, since the extra radius does not depend on the size of the circle.

Worked examples

• From the radius: r = 5 cm gives C = 2π × 5 ≈ 31.42 cm and A = π × 5² ≈ 78.54 cm².
• From the diameter: a round table 1.2 m across has r = 0.6 m and A = π × 0.36 ≈ 1.13 m².
• From the circumference: a tree trunk measured at 100 cm round has r = 100 ÷ 2π ≈ 15.92 cm and a cross-section of about 795.77 cm². A shortcut is A = C² ÷ 4π.
• From the area: to get a circle of 50 m², r = √(50 ÷ π) ≈ 3.99 m.

Why bigger circles win

Because the radius is squared, doubling it multiplies the area by four. Pizza is the classic test. A 12-inch pizza has an area of π × 6² ≈ 113.1 square inches, while two 8-inch pizzas have 2 × π × 4² ≈ 100.5, so the single large one is more food. A 16-inch pizza, only a third wider than a 12-inch, has about 201.1 square inches, roughly 1.78 times as much. Comparing sizes this way is a question of ratios, as Ratios explained: how to simplify, scale, and divide an amount in a given ratio shows, and the area calculator compares circles with rectangles and other shapes.

Slices and sectors

A sector is the slice between two radii. Its area is the fraction of the full turn it covers: a 90° quarter circle of radius 4 has area ¼ × π × 4² ≈ 12.57. In radians, the sector area is ½r²θ and the arc length is rθ. A 60° slice of the 5 cm circle has an arc of 5.24 cm and an area of 13.09 cm²; the calculator also gives the chord and the segment between chord and arc. The unit circle helps with the angles, and for 3D shapes built from circles, such as cylinders, cones and spheres, Surface area formulas: cubes, boxes, cylinders, spheres and cones takes the next step.