The area of a circle measures the flat region enclosed by its circular boundary. It is calculated from the radius: the distance from the centre to the boundary. If you know the diameter instead, halve it before using the radius formula.
Area from the radius or diameter
For radius r and diameter d, the equivalent formulas are:
The radius is squared, not doubled. With a radius of 5 cm, square 5 to obtain 25 and then multiply by π:
Working through the area of a circle with radius 5 cm
Find the area enclosed by a circle whose radius is 5 cm. The given length is already the radius, so it does not need to be halved.
Square the radius by multiplying 5 by 5. This is different from doubling 5.
Multiply the squared radius by π. Leave π in the expression to retain the exact area.
Evaluate the exact expression and round to two decimal places. Use the approximation sign for this rounded value.
The area is 25π square centimetres, approximately 78.54 square centimetres. An answer in centimetres alone would describe a length, not an area.
The exact answer is 25π square centimetres. The displayed decimal is rounded to two places. A diameter of 10 cm describes the same circle, so it must produce the same area.
Why does π appear in the area formula?
Imagine dividing a regular polygon inscribed in the circle into triangles from its centre. Its area equals half its perimeter multiplied by its apothem, the perpendicular distance from the centre to a side. As the number of sides increases, the perimeter approaches the circumference and the apothem approaches the radius.
In the limit, this gives half the circumference multiplied by the radius:
A finite polygon still has straight sides and does not equal the disk. This limiting argument explains the circle formula without treating an ordinary polygon as already circular.
Circle, disk and circumference
The circular boundary consists of points exactly r from the centre. The disk contains every point whose distance from the centre is at most r. Everyday English often uses circle for the whole figure, while disk makes the filled region explicit.
Area belongs to the filled region. Circumference is the length around its boundary. The boundary itself does not have the disk's area. Their formulas and units reflect that distinction:
For r = 5 cm, the circumference is 10π cm, while the area is 25π square centimetres. Neither the diameter nor the circumference is simply 2π: each depends on the size of the circle.
Units and rounding that change the answer
Convert lengths to the same unit before calculating. A diameter given in millimetres cannot be treated as a radius in centimetres. Because the formula squares a length, converting area involves the square of the length conversion factor. One square centimetre contains 100 square millimetres.
Doubling the radius multiplies the area by four but only doubles the circumference. This gives a useful reasonableness check. For a ring or a sector, the whole-circle formula needs an additional step, such as subtracting the inner disk or taking the appropriate fraction of the area. For a curved three-dimensional surface, see surface area of a sphere.
