Circle Calculator

Circle geometry is built on a single constant — π (pi) ≈ 3.14159 — and the relationship between a circle’s radius, diameter, circumference, and area. Knowing any one of these four values is enough to calculate all the others.

Fundamental relationships:

Diameter (d) = 2 × r

Circumference (C) = 2 × π × r = π × d

Area (A) = π × r²

Reverse formulas — solving from any known value:

From diameter: r = d ÷ 2

From circumference: r = C ÷ (2π) d = C ÷ π

From area: r = √(A ÷ π) d = 2 × √(A ÷ π)

Arc and sector formulas (with central angle θ in degrees):

Arc Length = (θ ÷ 360) × 2πr

Sector Area = (θ ÷ 360) × πr²

Worked example: A circular fountain has a diameter of 4.5 meters.

• Radius: 4.5 ÷ 2 = 2.25 m
• Circumference: π × 4.5 = 14.14 m (amount of edging needed)
• Area: π × 2.25² = 15.90 m² (surface area for a cover)

Now a sector example — a pizza slice cut to a 60° angle from a 14-inch (diameter) pizza:

• Radius = 7 in
• Arc length: (60 ÷ 360) × 2 × π × 7 = 7.33 inches
• Sector area: (60 ÷ 360) × π × 49 = 25.66 in²

Real-world uses of circle formulas:

• Landscaping: calculating sod or mulch for a circular bed
• Fencing: determining how much edging surrounds a circular pool
• Engineering: computing cross-sectional area of pipes and shafts
• Cooking: scaling a round cake recipe between pan sizes (area ratio)

Circle vs. sphere: This calculator handles flat 2D circles. For a 3D sphere, surface area = 4πr² and volume = (4/3)πr³.