Maths⏱ 5 min read
How to Calculate the Circumference and Area of a Circle
Circles appear everywhere in construction, engineering, and everyday life. Here's the essential formulas for circumference, area, arc length, and sector area — with worked examples.
The circle is one of the most fundamental shapes in geometry, and its formulas appear in construction (circular foundations, pipes), engineering (wheels, gears), and everyday calculations. All of them derive from π (pi ≈ 3.14159).
Key Measurements of a Circle
Radius (r): distance from centre to edge
Diameter (d): distance across the full circle = 2r
Circumference (C): the perimeter — the distance around the outside
Area (A): the space enclosed inside
π (pi) = 3.14159265... (irrational number, approximately 22/7)
Circumference
C = 2πr = πd
Using radius: C = 2 × 3.14159 × r
Using diameter: C = 3.14159 × d
Example: circle with radius 7cm
C = 2 × 3.14159 × 7 = 43.98 cm ≈ 44 cm
Example: wheel with diameter 65cm (26-inch bike wheel)
C = 3.14159 × 65 = 204.2 cm = 2.042 m per revolution
Area
A = πr²
Example: circular patio with radius 3m
A = 3.14159 × 3² = 3.14159 × 9 = 28.27 m²
Example: from diameter — first find r = d/2
Circular pond, diameter 4.5m → r = 2.25m
A = 3.14159 × 2.25² = 3.14159 × 5.0625 = 15.90 m²
Finding Radius or Diameter from Circumference or Area
From circumference:
r = C ÷ (2π)
d = C ÷ π
From area:
r = √(A ÷ π)
d = 2 × √(A ÷ π)
Example: circular room, area = 50 m². What diameter?
r = √(50 ÷ 3.14159) = √15.915 = 3.989 m
d = 2 × 3.989 = 7.98 m ≈ 8m diameter
Arc Length and Sector Area
An arc is a portion of the circumference; a sector is the "pie slice" shape enclosed by two radii and an arc.
Arc length = (θ ÷ 360) × 2πr [θ in degrees]
Sector area = (θ ÷ 360) × πr²
Example: sector with radius 10cm, angle 72°
Arc length = (72 ÷ 360) × 2π × 10 = 0.2 × 62.83 = 12.57 cm
Sector area = (72 ÷ 360) × π × 10² = 0.2 × 314.16 = 62.83 cm²
(Note: a 72° sector is exactly 1/5 of the circle,
so sector area = total area ÷ 5 ✓)
Annulus (Ring) Area
Annulus area = π(R² − r²)
R = outer radius, r = inner radius
Example: concrete ring footing, outer radius 4m, inner radius 3m
Area = π(4² − 3²) = π(16 − 9) = π × 7 = 21.99 m²
Practical Applications
ApplicationFormula NeededExample
Circular lawn areaA = πr²Fertiliser coverage
Pipe circumferenceC = πdLagging / insulation quantity
Bike wheel revolutionC = πdDistance per crank rotation
Round table clothC + overhangFabric length needed
Pizza slice areaSector areaPortion size comparison