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How to Calculate the Area and Circumference of a Circle
The area and circumference of a circle both come from one measurement. Learn the formulas, how radius, diameter, circumference and area connect, and work any one from another.
Updated 26 October 2026 · 2 min read
Every measurement of a circle — its area, circumference, diameter and radius — is locked to every other one by a single constant. Know any one of them and the rest follow. The star of the show is π (pi), roughly 3.14159, the fixed ratio of a circle's circumference to its diameter that holds for circles of every size.
The two formulas you need
Start with the radius (r), the distance from the centre to the edge. The two formulas that do almost all the work are:
Say a circle has a radius of 5 cm. Its area is π × 5² = π × 25 ≈ 78.54 cm², and its circumference is 2 × π × 5 ≈ 31.42 cm. Notice the area uses the radius squared while the circumference uses it directly — that squaring is why area grows so much faster than width as a circle gets bigger.
Radius, diameter, circumference, area — how they connect
- Diameter is twice the radius (d = 2r), and the radius is half the diameter.
- Circumference is π times the diameter, so you can reach the radius from a known circumference with r = C ÷ (2π).
- Area depends on r², so to go the other way you take a square root: r = √(Area ÷ π).
That last step is where circles meet square roots — if you know the area and want the radius, you have to undo the squaring. The circle calculator lets you enter whichever value you happen to have — radius, diameter, circumference or area — and instantly returns the other three, handling the π and the square root for you.
Why area quadruples when radius doubles
Because area is proportional to r², doubling the radius multiplies the area by four, not two. A pizza of twice the diameter gives you four times as much pizza. It's a handy sanity check whenever a problem seems to grow faster than you expect: with circles, the growth is in the square.
Where it shows up in everyday life
- Working out how much paint, turf or flooring a circular area needs.
- Comparing pizza or cake sizes by their true surface area rather than by diameter.
- Sizing pipes, tanks and planters, where the cross-section is a circle.
Circles are the gateway to a lot of geometry. Once the area formula feels natural, our guide to square roots covers the step that lets you run the calculation backwards, and the mean, median and mode walkthrough is a good next read if you're brushing up on the basics generally.
Frequently asked questions
- What's the difference between radius and diameter?
- The radius runs from the centre to the edge; the diameter runs all the way across through the centre. The diameter is exactly twice the radius, so if the radius is 5, the diameter is 10.
- Do I use π = 3.14 or something more precise?
- 3.14 is fine for rough work, but it introduces small errors. For accurate results use more digits of π (3.14159 or more). A calculator uses the full-precision value built into the computer, so its answers are as exact as the inputs you give.
- How do I find the radius if I only know the area?
- Rearrange the area formula: r = √(Area ÷ π). Divide the area by π, then take the square root. For an area of 50, that's √(50 ÷ 3.14159) ≈ √15.92 ≈ 3.99.