WorkedMathGlossary

Circle Calculator

Enter a radius below to instantly get area, circumference, and diameter.

Area
78.5398
Circumference
31.4159
Diameter
10
r = 5

Circle Formulas

Area = π × r². Circumference = 2 × π × r. Diameter = 2 × r.

A common mistake is plugging the diameter into the area formula in place of the radius. Because area depends on r², using the diameter by accident doesn't just double the result — it quadruples it, since (2r)² = 4r². Always halve the diameter to get the radius before squaring.

Worked example

A circle with radius 5.

Area: π × 5² = π × 25 ≈ 78.54

Circumference: 2 × π × 5 ≈ 31.42

A second example, with radius 10.

Area: π × 10² = π × 100 ≈ 314.16

Circumference: 2 × π × 10 ≈ 62.83

Keep going

  • A circle rotated around its diameter sweeps out a sphere; the sphere calculator extends this same radius into 3D volume and surface area. Sphere Calculator
  • A circle's circumference is the full loop; the arc length calculator finds just a portion of it, from a radius and a central angle. Arc Length Calculator
  • Every circle's circumference and arc length scale with its radius; the unit circle calculator fixes the radius at exactly 1, which is what makes an angle's coordinates equal cos and sin directly. Unit Circle Calculator

Frequently Asked Questions

How do you find the area of a circle?

Area = π × r², where r is the radius. Multiply the radius by itself, then by π (about 3.14159).

How do you find the circumference of a circle?

Circumference = 2 × π × r, or equivalently π × d, where d is the diameter.

What's the difference between radius and diameter?

The radius is the distance from the center to the edge. The diameter is the full distance across the circle through the center — exactly twice the radius.

Why is the area formula π × r² and not something else?

It comes from slicing the circle into many thin pie-shaped wedges and laying them out in alternating directions to approximate a parallelogram. The wedges' straight edges are each a radius (r), and the total length across them works out to half the circumference (πr). Multiplying that length by the height (r) gives πr × r = πr² — and the approximation becomes exact as the wedges get infinitely thin.

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