Area of a Circle Calculator
Solve any circle measurement in both directions: enter the radius, diameter, circumference, or area and get all four in return. Switch to sector mode for a "pizza slice" area by angle, or annulus mode for a ring between two circles. Every result comes with a step-by-step worked solution and an exact answer in terms of pi.
Formula
Worked example
Radius 5 m: A = pi x 5^2 = 78.5398 m^2, C = 2pi x 5 = 31.4159 m, diameter = 10 m, exact = 25pi m^2. Sector 90 deg, r = 5 m: sector area = 0.5 x 25 x (pi/2) = 19.6350 m^2, arc = 7.8540 m. Annulus R = 8 m, r = 5 m: area = pi(64 - 25) = 39pi = 122.5221 m^2.
How the area of a circle is calculated
The area of a circle is the amount of flat space enclosed inside its boundary. The formula is A = pi r squared, where r is the radius and pi is approximately 3.14159. Because the radius is squared, area grows much faster than the linear dimensions: a circle twice as wide encloses four times as much area, not twice. If you know the diameter instead of the radius, halve it first, or use the equivalent form A = pi d squared divided by four directly. The circumference (perimeter of the circle) is 2 pi r, which is also pi times the diameter. These three measurements are tightly linked: knowing any one of them is enough to find all the others.
Reverse-solve: find the radius from area or circumference
Sometimes you measure the area or perimeter of a circular space and need to work backwards to the radius. This calculator handles both reverse cases. If you know the area A, the radius is the square root of (A divided by pi). If you know the circumference C, the radius is C divided by (2 pi). Switch the "I know the" dropdown to Area or Circumference and enter your measured value to use these modes. The same four outputs (radius, diameter, area, circumference) are returned so you have the full picture.
Sector area: the "pizza slice"
A sector is the region bounded by two radii and the arc between them, the shape of a pizza slice or a pie wedge. Its area is a fraction of the full circle proportional to the central angle: A_sector = (theta divided by 360) times pi r squared, where theta is the angle in degrees. The equivalent formula using radians is 0.5 r squared times theta_rad. The arc length (the curved edge of the slice) is r times the angle in radians, or equivalently (theta divided by 360) times the full circumference. Switch this calculator to Sector mode, enter the radius and the central angle, and both the sector area and arc length are returned together.
Annulus area: the ring between two circles
An annulus is the region between two concentric circles sharing the same centre. It looks like a washer, a pipe cross-section, or a circular running track. Its area is simply the outer circle area minus the inner circle area: pi times (R squared minus r squared), where R is the outer radius and r is the inner radius. This can also be written as pi times (R plus r) times (R minus r), which is useful for quick mental checks. Common applications include calculating the cross-sectional area of hollow tubes, the material in a circular picture frame, or the area of a ring-shaped garden bed.
Unit switch: metric and imperial
All inputs support both metric (metres, centimetres, millimetres, kilometres) and imperial (feet, inches, yards, miles) units. The unit is chosen per-field with the unit-number input, and the calculator converts everything to metres internally before computing, so you can mix units freely. Results are returned in SI (square metres for area, metres for length), and you can scale them to any unit you need by multiplying by the appropriate factor. For square unit conversions, remember that 1 foot squared = 0.0929 m squared, and 1 inch squared = 0.000645 m squared.
Why pi appears in every circle formula
Pi (approximately 3.14159) is the ratio of any circle's circumference to its diameter, and it appears in every formula involving circles because circles are defined by that constant ratio. It is irrational (its decimal expansion never ends or repeats) and transcendental (it is not the root of any polynomial with rational coefficients). For practical calculations, using pi to 5 decimal places (3.14159) gives answers accurate to within about 0.001%, which is far more precise than most real-world measurements. This calculator uses JavaScript's Math.PI constant (approximately 15 significant figures) and rounds only at the display stage.
Circle measurements reference table
| Radius | Diameter | Area (pi r^2) | Circumference (2pi r) | 90-deg sector area | 180-deg sector area |
|---|---|---|---|---|---|
| 1 | 2 | 3.1416 | 6.2832 | 0.7854 | 1.5708 |
| 2 | 4 | 12.5664 | 12.5664 | 3.1416 | 6.2832 |
| 5 | 10 | 78.5398 | 31.4159 | 19.6350 | 39.2699 |
| 10 | 20 | 314.1593 | 62.8319 | 78.5398 | 157.0796 |
| 25 | 50 | 1963.4954 | 157.0796 | 490.8739 | 981.7477 |
| 50 | 100 | 7853.9816 | 314.1593 | 1963.4954 | 3926.9908 |
Area, circumference, and sector area for common radii (all values in the given unit, using that unit squared for area).
Frequently asked questions
What is the formula for the area of a circle?
The area of a circle is A = pi r squared, where r is the radius and pi is approximately 3.14159. If you know the diameter d instead, the equivalent formula is A = pi d squared divided by 4, since r = d divided by 2. The result is always in square units of whatever length unit you used.
How do I find the area if I only know the circumference?
Divide the circumference by (2 pi) to get the radius, then apply A = pi r squared. Alternatively use the direct formula A = C squared divided by (4 pi). For example, a circumference of 31.4159 m gives r = 31.4159 / (2 pi) = 5 m, so A = pi x 25 = 78.5398 m squared. Select "Circumference" in the "I know the" dropdown and this calculator does it automatically.
How do I find the radius from a known area (reverse-solve)?
Take the square root of (area divided by pi). For example, if the area is 78.5398 m squared, the radius is sqrt(78.5398 / pi) = sqrt(25) = 5 m. Select "Area (reverse-solve)" from the input mode dropdown in this calculator and enter the area value.
What is a sector of a circle and how is its area calculated?
A sector is a "pizza slice" region bounded by two radii and the arc between them. Its area is a fraction of the full circle based on the central angle: A_sector = (angle_degrees / 360) x pi r squared. In radians the formula is 0.5 r squared x angle_radians. Switch this calculator to Sector mode and enter the radius and the central angle in degrees.
What is an annulus and what is its area formula?
An annulus is the ring-shaped region between two concentric circles. Its area equals the outer circle area minus the inner circle area: A = pi (R squared minus r squared), where R is the outer radius and r is the inner radius. This formula applies directly to pipe cross-sections, washers, circular tracks, and hollow cylinders. Switch this calculator to Annulus mode to compute it.
What are the units of the area output?
Area is in square metres (m squared) when the input radius is in metres, square centimetres when the radius is in centimetres, and so on. The unit is always the square of whatever length unit you enter. This calculator converts all inputs to metres internally and returns the area in square metres; multiply by 10,000 to convert to cm squared, or by 10.7639 to convert to square feet.