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Circle Calculator

Enter any one measurement of a circle — radius, diameter, circumference, or area — and this calculator computes all the others. Formulas and step-by-step work are shown below each result.

Select which measurement you know, then enter its value:

Results

Radius (r):5
Diameter (d = 2r):10
Circumference (C = 2πr):31.41592654
Area (A = πr²):78.53981634
Step-by-step:
Given: radius = 5

Step 1 — Find radius:
  r = 5 (given)

Step 2 — Diameter:
  d = 2r = 2 × 5 = 10

Step 3 — Circumference:
  C = 2πr = 2 × 3.14159265 × 5
  C = 31.41592654

Step 4 — Area:
  A = πr² = 3.14159265 × 5²
  A = 3.14159265 × 25
  A = 78.53981634

Circle Formulas

All circle formulas derive from one number: the radius r, the distance from the center to the edge. The constant π ≈ 3.14159265 links linear measurements (radius, diameter) to curved measurements (circumference) and area.

Measurement Formula Example (r = 5)
Diameter d = 2r d = 2 × 5 = 10
Circumference C = 2πr = πd C = 2 × π × 5 ≈ 31.416
Area A = πr² A = π × 25 ≈ 78.540

Finding Any Measurement from Any Other

If you know any single measurement, you can find all others by rearranging the formulas:

Known Radius Diameter Circumference Area
Radius (r) — 2r 2πr πr²
Diameter (d) d/2 — πd πd²/4
Circumference (C) C/(2π) C/π — C²/(4π)
Area (A) √(A/π) 2√(A/π) 2√(πA) —

What Is Pi (π)?

Pi (π) is the ratio of any circle's circumference to its diameter: π = C / d. This ratio is exactly the same for every circle, from a coin to a planet. Pi is irrational (cannot be expressed as a fraction) and transcendental (not a root of any polynomial with rational coefficients).

Approximations used in practice:

For most engineering and everyday calculations, π ≈ 3.14159 is more than sufficient.

Arc Length and Sector Area

A sector is a "pie slice" of a circle with central angle θ:

For a semicircle (θ = 180°): arc = πr, area = πr²/2.
For a quarter circle (θ = 90°): arc = πr/2, area = πr²/4.

Real-World Circle Calculations

Frequently Asked Questions

What is the formula for the circumference of a circle?

Circumference C = 2πr, where r is the radius. You can also write it as C = πd, where d is the diameter. Since π ≈ 3.14159, a circle with radius 5 has circumference 2 × 3.14159 × 5 ≈ 31.416. Circumference is the distance around the circle — its perimeter.

What is the formula for the area of a circle?

Area A = πr², where r is the radius. A circle with radius 5 has area π × 25 ≈ 78.540. Area grows with the square of the radius, so doubling the radius quadruples the area. This is why a 16-inch pizza has four times the area of an 8-inch pizza.

What is pi (π) and why does it appear in circle formulas?

Pi (π) is the ratio of a circle's circumference to its diameter: π = C/d ≈ 3.14159265. This ratio is constant for every circle, regardless of size. Pi is irrational (its decimal never repeats) and transcendental. It appears in circle formulas because the relationship between linear dimensions (radius, diameter) and curved dimensions (circumference) requires this universal constant.

How do you find the radius from the area?

Rearrange A = πr² to solve for r: r = √(A/π). For example, if area = 50, then r = √(50/π) = √(50/3.14159) = √15.915 ≈ 3.989. You can then find diameter = 2r ≈ 7.978 and circumference = 2πr ≈ 25.066.

What is a sector and how is its area calculated?

A sector is a "pie slice" of a circle — the region between two radii and the arc connecting them. Sector area = (θ/360°) × πr², where θ is the central angle in degrees. Arc length = (θ/360°) × 2πr. For a semicircle (θ = 180°), sector area = πr²/2. For a quarter circle (θ = 90°), sector area = πr²/4.

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