Mathematics · Geometry

Calculate the area of a circle

One length is enough: the radius, or the diameter (halved). You get area πr², circumference 2πr, and both r and d as a check.

Calculate

How the calculator works

If you give the diameter, r = d/2. Area is π × r × r. Circumference is 2πr. π is not 22/7 or 3.14: it is the engine’s Math.PI, plenty for a plan, a yard, or a class exercise. Zero or negative lengths are refused.

Formula and method

A = π r²    and    C = 2 π r    with    d = 2 r

For r = 1, A = π ≈ 3.14159265 u² and P = 2π ≈ 6.28318531 u. For diameter 10, r = 5, A = 25π ≈ 78.539816.

A single numeric input, interpreted as radius or diameter. No ellipse, no sector, no annulus. MathWorld’s disk formulas are the reference; the implementation does not invent another constant besides Math.PI.

Worked example

Unit radius

known = radius, length = 1

r = 1, d = 2.
Area = π × 1² = π.
Circumference = 2π.

Area ≈ 3.141593 u², circumference ≈ 6.283185 u.

Diameter 10

known = diameter, length = 10

r = 10 / 2 = 5.
Area = π × 25 = 25π ≈ 78.539816.

Area ≈ 78.539816 square units, circumference 10π ≈ 31.415927.

Input notes

Known measurement: Tells the engine whether the number is r or d.
Length: Must be > 0. Diameter 10 is not radius 10.

Assumptions and limits

Assumptions

A true circle in the plane.
One length unit; area is that unit squared.

Limits

Not a sphere (surface 4πr²) and not a cylinder.
π approximation is floating-point, not a symbolic π.

How to read the result

If r is in feet, area is in square feet. A diameter typed as if it were a radius inflates area by four. For a disk of mulch, convert inches to meters if you then use the mulch volume tool, which wants meters.

Common mistakes

Using diameter in πr² without halving.
If the tape reading is a diameter, choose diameter mode or divide by 2 first.
Reporting circumference as area.
2πr is a length. πr² is an area.

Methodology · Sources