The area of a circle is A = πr², and its circumference is C = 2πr — this circle calculator applies both formulas instantly from whichever measurement you already have. Enter the radius, diameter, circumference, or area, and it solves for all the other values at once.
The constant π (pi) is approximately 3.14159 and is the same for every circle: it is the ratio of any circle’s circumference to its diameter. That single fact links all four measurements together, which is why knowing just one of them is enough to determine the entire circle.
Circle Formulas: Area, Circumference, Radius, Diameter
Every circle measurement can be derived from the radius r:
- Area: A = πr²
- Circumference: C = 2πr (equivalently C = πd)
- Diameter: d = 2r
Working backwards from the other measurements:
- From the diameter: r = d ÷ 2
- From the circumference: r = C ÷ 2π
- From the area: r = √(A ÷ π)
Note the difference in units: circumference is a length (measured in units like cm or ft), while area covers a surface (measured in square units like cm² or ft²). Doubling the radius doubles the circumference but quadruples the area, because the radius is squared in the area formula.
Worked Example 1: Area of a Circle with Radius 6
Find the area of a circle with radius r = 6 cm.
- Apply the formula: A = πr² = π × 6² = π × 36.
- Multiply: A = 36π ≈ 113.10 cm².
The circumference of the same circle is C = 2πr = 2 × π × 6 = 12π ≈ 37.70 cm, and the diameter is d = 2 × 6 = 12 cm.
A common mistake is squaring the diameter instead of the radius. If you are given the diameter (12 cm here), halve it first: r = 6, then A = π × 36 — not π × 144, which would be four times too large.
Worked Example 2: From Circumference Back to Area
A circular garden has a circumference of 31.4159 m. What is its area?
- Find the radius: r = C ÷ 2π = 31.4159 ÷ 6.28319 ≈ 5 m.
- Apply the area formula: A = πr² = π × 25 ≈ 78.54 m².
This two-step conversion works in either direction, and it is exactly what the calculator does internally: whatever value you supply is first converted to the radius, and everything else follows from r.
Quick reference for common radii: r = 1 → A ≈ 3.14; r = 2 → A ≈ 12.57; r = 3 → A ≈ 28.27; r = 5 → A ≈ 78.54; r = 10 → A ≈ 314.16.
Frequently Asked Questions
What is the formula for the area of a circle?
The area of a circle is A = πr², where r is the radius and π ≈ 3.14159. Square the radius, then multiply by pi. If you only know the diameter, divide it by 2 first to get the radius. For example, a circle with radius 4 has area π × 16 ≈ 50.27 square units.
How do you find the circumference of a circle?
Circumference is C = 2πr, or equivalently C = πd using the diameter. Multiply the radius by 2, then by π ≈ 3.14159. A circle with radius 7 has circumference 2 × π × 7 ≈ 43.98 units. Circumference is the distance around the circle — a length, not an area.
What is the difference between radius and diameter?
The radius is the distance from the center of the circle to its edge, while the diameter is the full distance across the circle through the center. The diameter is always exactly twice the radius: d = 2r. So a circle with a 10 cm diameter has a 5 cm radius.
How do I find the radius from the area?
Rearrange A = πr² to get r = √(A ÷ π). Divide the area by pi, then take the square root. For example, if the area is 100 square units, r = √(100 ÷ 3.14159) = √31.83 ≈ 5.64 units. From there you can compute the diameter and circumference.
What is π (pi) and why does it appear in circle formulas?
Pi is the ratio of any circle’s circumference to its diameter — about 3.14159, the same for every circle regardless of size. It is an irrational number, so its decimals never repeat or end. Because circumference and diameter are locked in this fixed ratio, π shows up in every formula involving circles, including area (πr²).