Circles appear everywhere — wheels, pipes, circular rooms, pizza, planets. Two measurements define every circle completely: the circumference (the distance around the edge) and the area (the space inside). Both follow directly from a single value: the radius.
Key Terms
Radius (r): The distance from the centre of the circle to any point on its edge. This is the fundamental measurement — all circle formulas use it.
Diameter (d): The distance across the circle through the centre. Always exactly twice the radius: d = 2r.
Circumference (C): The perimeter of the circle — the total distance around the outside edge.
Area (A): The amount of two-dimensional space enclosed by the circle.
π (pi): The ratio of any circle's circumference to its diameter. It is irrational (never-ending, never-repeating) and approximately equal to 3.14159265...
Circumference Formula
C = 2πr or equivalently C = πd
Example: A circle with radius 5 cm
C = 2 × π × 5 = 10π ≈ 31.42 cm
In terms of diameter: If given the diameter directly:
C = π × d = π × 10 = 10π ≈ 31.42 cm
Both give the same answer — choose whichever measurement you have.
Area Formula
A = πr²
Example: Same circle with radius 5 cm
A = π × 5² = 25π ≈ 78.54 cm²
Note: area is always in square units (cm², m², in²). Circumference is in linear units (cm, m, in).
Working Backwards from Circumference or Area
Sometimes you know the circumference or area and need to find the radius.
Radius from circumference:
r = C / (2π)
Radius from area:
r = √(A / π)
Diameter from circumference:
d = C / π
Example: A circular field has circumference 150 m. What is its area?
Step 1: Find radius
r = 150 / (2π) = 150 / 6.2832 = 23.87 m
Step 2: Find area
A = π × 23.87² = π × 569.8 ≈ 1,790 m²
Common Worked Examples
Circular pipe cross-section
A pipe has internal diameter 40 mm. What is the cross-sectional area?
r = 40 / 2 = 20 mm
A = π × 20² = 400π ≈ 1,257 mm²
This matters for flow rate calculations — the area determines how much fluid can pass through.
Running track
A circular running track has radius 40 m. How far is one lap?
C = 2π × 40 = 80π ≈ 251.3 m
(Standard 400 m tracks are actually oval, not circular — but this shows the principle.)
Pizza size comparison
Is a 14-inch pizza worth more than two 10-inch pizzas?
14-inch pizza:
A = π × 7² = 49π ≈ 153.9 in²
Two 10-inch pizzas:
A = 2 × π × 5² = 2 × 25π = 50π ≈ 157.1 in²
Two 10-inch pizzas give very slightly more pizza — but only if the price is comparable.
Sectors and Arcs
A sector is a "slice" of a circle (like a pie slice), defined by a central angle θ.
Arc length (the curved edge of the sector):
Arc = (θ / 360) × 2πr [degrees]
Arc = θr [radians]
Sector area:
Sector area = (θ / 360) × πr² [degrees]
Sector area = ½r²θ [radians]
Example: Sector with radius 8 cm and central angle 45°
Arc length = (45 / 360) × 2π × 8 = (1/8) × 16π = 2π ≈ 6.28 cm
Sector area = (45 / 360) × π × 64 = (1/8) × 64π = 8π ≈ 25.13 cm²
Annulus (Ring Shape)
An annulus is the region between two concentric circles with radii R (outer) and r (inner).
Annulus area = π(R² − r²) = π(R + r)(R − r)
Example: A circular border with outer radius 10 m and inner radius 7 m:
Area = π(10² − 7²) = π(100 − 49) = 51π ≈ 160.2 m²
Summary of Formulas
| Measurement | Formula | |-------------|---------| | Circumference | C = 2πr = πd | | Area | A = πr² | | Radius from C | r = C / (2π) | | Radius from A | r = √(A/π) | | Arc length (degrees) | Arc = (θ/360) × 2πr | | Sector area (degrees) | A = (θ/360) × πr² | | Annulus area | A = π(R² − r²) |
Use our Circle Calculator to compute any circle measurement — enter any one value and get all the others instantly.