Circle circumference and area
Every circle calculation involves the constant π (pi). Two formulas cover all circle questions: one for the distance around the edge and one for the space inside. Once you know the radius or diameter, both are a single multiplication away.
Core formulas
Circumference: C = 2πr = πd
Area: A = πr²
where r = radius, d = diameter = 2r, and π ≈ 3.14159
Parts of a circle
The radius (r) is the distance from the center of the circle to any point on its edge. The diameter (d) is the distance across the circle through the center — always exactly twice the radius. The circumference (C) is the total distance around the outside edge.
Any two of these three measurements — plus π — let you find the rest.
Understanding π
π (pi) is the ratio of any circle’s circumference to its diameter. It is the same for every circle of any size: divide the circumference by the diameter and you always get π.
π is an irrational number — its decimal expansion never repeats or terminates. For most problems, use π ≈ 3.14159 and round your final answer to 2 decimal places.
From radius: two formulas
C = 2 × π × r
A = π × r × r
From diameter: convert first
C = π × d
A = π × (d ÷ 2)²
Quick reference
π ≈ 3.14159 | π ≈ 3.14 (2 d.p.) | π ≈ 22/7
Radius vs diameter: always check
Many circle mistakes come from confusing radius and diameter. Before applying any formula, check whether the problem gives you a radius (center to edge) or a diameter (all the way across).
- If the problem says “radius of 5”, then r = 5 and d = 10.
- If the problem says “diameter of 10”, then d = 10 and r = 5.
- The area formula uses r only. If you are given d, halve it before squaring.
- The circumference formula can use either: C = 2πr or C = πd — choose the one that matches your given value.
Worked examples
Example 1: circumference from radius
A circle has a radius of 5 cm. Find the circumference. Use π ≈ 3.14159.
C = 2 × π × r = 2 × 3.14159 × 5 = 31.42 cm
Multiply radius by 2, then by π. Or: multiply by 2π ≈ 6.28318 in one step.
Example 2: area from radius
A circle has a radius of 7 m. Find the area. Use π ≈ 3.14159.
A = π × r² = 3.14159 × 7² = 3.14159 × 49 ≈ 153.94 m²
Square the radius first (7² = 49), then multiply by π. Do NOT square π — only r is squared.
Example 3: circumference from diameter
A circle has a diameter of 12 ft. Find the circumference.
C = π × d = 3.14159 × 12 ≈ 37.70 ft
When given a diameter, use C = πd directly — no need to halve first for circumference.
Example 4: area from diameter
A circle has a diameter of 10 in. Find the area.
First find r: r = d ÷ 2 = 10 ÷ 2 = 5
A = π × r² = 3.14159 × 25 ≈ 78.54 in²
For area, always convert diameter to radius first. Squaring the diameter (100 instead of 25) would give an answer four times too large.
Handy reference: common radius values
| Radius (r) | Diameter (d) | Circumference (C ≈) | Area (A ≈) |
|---|---|---|---|
| 1 | 2 | 6.28 | 3.14 |
| 2 | 4 | 12.57 | 12.57 |
| 3 | 6 | 18.85 | 28.27 |
| 4 | 8 | 25.13 | 50.27 |
| 5 | 10 | 31.42 | 78.54 |
| 6 | 12 | 37.70 | 113.10 |
| 7 | 14 | 43.98 | 153.94 |
| 8 | 16 | 50.27 | 201.06 |
| 9 | 18 | 56.55 | 254.47 |
| 10 | 20 | 62.83 | 314.16 |
Values are rounded to 2 decimal places. Notice that when r = 2, circumference and area happen to be equal (both 12.57) — this is a coincidence at that specific size.
Refreshable sample
Try one circle question
Work through the problem before checking. Use π ≈ 3.14159 and round to 2 decimal places. The widget accepts any answer within 0.5 of the correct value to allow for rounding differences.
Common mistakes to avoid
- Squaring π instead of r. In A = πr², only r is squared. Writing (πr)² or π²r gives completely wrong answers.
- Using diameter where radius is required. The area formula needs the radius. If you plug the diameter straight into A = πr² without halving it first, you get an answer four times too large.
- Forgetting to multiply by 2 in the circumference formula. C = πr (only one radius) gives half the correct circumference. The formula is C = 2πr.
- Rounding π too early. Round only at the final step. Rounding π to 3.1 mid-calculation introduces larger errors than rounding 3.14159 at the end.
- Wrong units. Circumference is a length (cm, m, in). Area is a square measurement (cm², m², in²). Never label a circumference answer as “square units.”