The radius of the circle is:
Table of Contents
Radius from area, instantly
Enter the area of a circle and this calculator returns the radius using r = √(A/π). Useful when you know the surface area of a circular plot, pipe, lid or coin but need the distance from the centre to the edge.
The formula
The radius is the square root of the area divided by pi:
r = √(A / π)
- r — the radius, in the same length unit as the square root of the area.
- A — the area, in square units (e.g. m², in²).
- π — pi, approximately 3.14159.
This is the inverse of A = πr². Divide by pi to undo the multiplication, then square-root to undo the squaring.
Worked example
Using the default area of 314 square units:
- Divide by pi: 314 / 3.14159 = 99.9493...
- Take the square root: √99.9493 = 9.9975...
So r ≈ 10 units. (314 is a rounded version of π × 100, so the exact answer for an area of exactly π × 100 is 10.)
Reverse formulas at a glance
| If you know… | Then the radius is… |
|---|---|
| Area, A | r = √(A / π) |
| Circumference, C | r = C / (2π) |
| Diameter, d | r = d / 2 |
| Sphere volume, V | r = ∛(3V / 4π) |
| Sphere surface area, S | r = √(S / 4π) |
Reference table
Radius for common areas, using π ≈ 3.14159:
| Area | Radius | Diameter |
|---|---|---|
| 3.14 | 1.00 | 2.00 |
| 12.57 | 2.00 | 4.00 |
| 78.54 | 5.00 | 10.00 |
| 314.16 | 10.00 | 20.00 |
| 1,963.50 | 25.00 | 50.00 |
| 7,853.98 | 50.00 | 100.00 |
Common applications
- Tank sizing. A water tank’s cross-sectional area is given by its capacity per metre of height — reverse-solve for the radius to spec pipework and fittings.
- Garden design. Sold a topsoil order by area? Divide by pi and square-root to figure out the radius of the round bed it fills.
- Sports markings. A throwing-circle in athletics is defined by area or diameter, but you mark it with a radius from the centre.
- Astronomy. Given the surface area of a planetary disk as seen from Earth, the formula recovers the body’s apparent radius.
Limitations & gotchas
- The area must be in square units. A common mistake is to enter a circumference value — that gives a wildly wrong radius.
- The formula assumes a true circle. For an ellipse you would need the semi-major and semi-minor axes (a, b) instead.
- Negative or zero inputs return non-physical results. Real circles have positive area.
- π is irrational; results are rounded for display. Internal precision is ~15 significant digits.
Sources & references
- Weisstein, Eric W. “Circle.” Wolfram MathWorld.
- Britannica, “circle (mathematics).” Encyclopaedia Britannica.
- Britannica, “pi (mathematics).” Encyclopaedia Britannica.
FAQs
Use r = √(A / π). Divide the area by pi, then take the square root. This is the inverse of A = πr². For example, an area of 314 gives r = √(314 / 3.14159) = √100 = 10.
Use r = C / (2π). Divide the circumference by 2π (about 6.2832). For example, a circumference of 62.83 gives r = 62.83 / 6.2832 ≈ 10. This is the inverse of C = 2πr.
The diameter is twice the radius: d = 2r, so r = d/2. The diameter is the longest straight line that fits inside the circle and always passes through the centre. If you measure across a coin or pipe with calipers you’ve measured the diameter; halve it to get the radius.
No. The radius is a distance, so it’s a non-negative real number. A radius of zero collapses the circle to a single point with no area and no circumference, which is a degenerate case rather than a useful one.
Because area depends on the square of the radius (A = πr²). To recover r from A, you have to undo the squaring — that’s what the square root does. The same is true for any inverse-of-power relationship: cube volumes invert with a cube root, and so on.
If the area was in square metres, the radius is in metres. If the area was in square inches, the radius is in inches. The square root strips one power of the unit: m² → m, in² → in. Always work in consistent units throughout.