The area of a circle is π times the radius squared. If you know the diameter (twice the radius), divide by 2 first. The classic value of π — 3.14159… — has been computed to trillions of digits, but for engineering purposes 3.14159 (or 3.1416) is more than enough.

How it works — the formula

A = π · r²        (radius)
A = π · d² / 4    (diameter)

The area of a circle is π times the square of its radius. The π constant is the ratio of any circle's circumference to its diameter, transcendental and irrational; π ≈ 3.14159265358979… The formula can be derived by integrating 2πr dr from 0 to R or by treating the disk as the limit of a regular polygon's area as the side count tends to infinity (Archimedes, ~250 BC).

Sources: NIST DLMF §1.10(i) — Special constants (π definition) · NIST CODATA — Recommended values: pi · Wolfram MathWorld — Disk area derivation

Worked examples

Example 1

Radius given

Inputs:: r = 5
Output:: A = π · 25 ≈ 78.5398

Example 2

Diameter given

Inputs:: d = 10
Output:: A = π · 100 / 4 ≈ 78.5398 (matches r = 5)

Example 3

Real-world: pizza area

Inputs:: r = 14 in (28-inch pizza diameter)
Output:: A ≈ 615.75 in² — twice the area of a 20-inch pizza

Limitations

Formula assumes a flat (Euclidean) plane; for the surface of a sphere or other curved manifold, use the relevant intrinsic geometry formula.
π is irrational, so any decimal area is a truncation; carry enough digits for the precision your downstream use requires.
Negative or zero radius input produces zero or a meaningless value; this calculator clamps to zero.
A real disk's area depends on whether you are measuring the outer boundary, painted surface, or projected outline — assumed identical here.

Area is exact within floating-point precision (~15–16 significant digits) using the IEEE 754 representation of π.

Frequently asked

What is π?+

The ratio of any circle's circumference to its diameter — approximately 3.14159265. It's irrational (doesn't terminate or repeat) and transcendental (not the root of any polynomial with rational coefficients).

How do I compute the area if I only know the circumference?+

Divide circumference by 2π to get radius, then plug into A = πr². Or use the identity A = C²/(4π).

How precise is the answer?+

JavaScript's Math.PI is accurate to ~15 significant digits, more than enough for any practical use.

How does this differ from area of an ellipse?+

A circle is a special ellipse where both semi-axes are equal. A = πa·b = πr·r = πr².

Can I use this for sectors or arcs?+

No — use the area-sector calculator for slices of a circle.