Circle Calculator

Every formula, in every direction

The calculator first converts whatever you enter into the radius, then computes the rest.

The formulas are the standard ones taught in OpenStax Prealgebra: C = 2πr, C = πd and A = πr². The calculator uses JavaScript's built-in value of π (3.141592653589793) rather than 3.14 or 22/7, so its answers differ slightly from textbook answers that use those approximations. For a 10-inch radius, 3.14 gives 314 sq in, while full π gives 314.159 sq in.

How much precision in π do you need?

Very little. NASA's Jet Propulsion Laboratory uses π to 15 decimal places (3.141592653589793) for its most precise interplanetary navigation. At that precision, the circumference of a circle about 40 billion miles across (roughly 125 billion miles around) would be off by about 1.5 cm. For homework, 3.14 or 3.1416 is usually what the problem expects; for building and DIY work, the difference between 3.14 and the full value is far smaller than the accuracy of a tape measure.

Area grows with the square of the size

Doubling the diameter doubles the circumference but quadruples the area. Euclid stated this in Elements XII.2: "Circles are to one another as the squares on their diameters."

That is why one 16-inch pizza (201 sq in) has almost as much as two 12-inch pizzas (226 sq in), and one 18-inch pizza (254 sq in) has more than two 12-inch pizzas. Divide the price by the area to compare cost per square inch.

U.S. units: square feet, square yards, acres

Area units are squared, so conversion factors are squared too:

• 1 sq ft = 144 sq in (12 × 12)
• 1 sq yd = 9 sq ft (3 × 3)
• 1 acre = 43,560 sq ft

A circular patio 14 ft across has an area of π × 7² ≈ 153.94 sq ft, or about 17.1 sq yd. A center-pivot irrigation circle with a 1,320-ft radius (a quarter mile) covers π × 1,320² ≈ 5,473,911 sq ft, which is about 125.7 acres. For materials sold by the cubic yard (concrete, mulch, gravel), multiply the area in square feet by the depth in feet and divide by 27.

Common mistakes

• Using the diameter in πr². If you plug the diameter in as r, the area comes out four times too large. A 14-inch pizza is not 615.75 sq in; it is 153.94 sq in.
• Squaring after multiplying by π. πr² means π × (r²), not (πr)².
• Mixing units. A radius in inches gives square inches. To get square feet, either convert the radius to feet first or divide the square-inch result by 144, not by 12.
• Confusing circumference and area. Circumference is a length (fence, edging, trim). Area is a surface (sod, paint, pavers).
• Treating a cylinder as a circle. For a round tank, pipe or column you need volume, πr² × height. The circle area here is only the base.

Practical uses

Edging or fencing around a round bed or pool uses the circumference. Sod, seed, paint, tablecloth fabric and pavers use the area. A tree's trunk diameter is commonly estimated by measuring around it at chest height and dividing by π, which is exactly the "circumference" mode. Pipe and duct sizing uses the cross-sectional area, πd² ÷ 4.

Semicircles, quarter circles, sectors and rings

Many real shapes are parts of a circle. Find the full-circle values here, then adjust:

A semicircular window 4 ft wide has r = 2 ft, so its glass area is π × 4 ÷ 2 ≈ 6.28 sq ft and its frame is π × 2 + 4 ≈ 10.28 ft. A 3-ft gravel path around a round pool 24 ft across is a ring with r = 12 ft and R = 15 ft: π × (225 − 144) = 81π ≈ 254.47 sq ft. A ring's area depends only on the difference of the squared radii, so widening the path from 3 ft to 4 ft (R = 16 ft) raises it to π × (256 − 144) = 112π ≈ 351.86 sq ft, an increase of almost 40% for one extra foot of width.