Circle Calculator

Radius · Diameter · Circumference · Area · Arcs · Equations

A = πr² · C = 2πr

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⭕ Enter one circle measurement, an arc angle, or an equation to see every related value with steps.

What Is a Circle? Radius, Diameter, Circumference, and Area

A circle is the set of points in a plane that are all the same distance from a center point. That distance is the radius (r). The diameter (d) is the distance straight across through the center, twice the radius. The circumference (C) is the distance around the circle, its perimeter. The area (A) is the flat surface enclosed inside it.

The ratio of any circle's circumference to its diameter is the constant π (pi), about 3.14159. Because all four measurements depend on the radius, knowing any one of them is enough to find the other three. This circle calculator does exactly that, and also covers parts of a circle (arcs, sectors, chords, and segments) and circle equations on a coordinate plane.

How to Use the Circle Calculator

Measurements: choose which value you know (radius, diameter, circumference, or area), type it, and pick a unit. The radius, diameter, circumference, and area appear instantly with the formula steps.
Arc and sector: enter the radius and the central angle in degrees or radians. You get the arc length, sector area, chord length, segment area, segment height, and sector perimeter.
Equation: enter a center and radius to get the standard and general equations, or enter D, E, and F from x² + y² + Dx + Ey + F = 0 to recover the center and radius. The calculator also lists the x and y intercepts and plots the circle.

Results use the full value of π and are rounded only for display. Area results are in square units of the length unit you choose.

Circle Formulas

Find	From r	From another value
Diameter	d = 2r	d = C ÷ π
Circumference	C = 2πr	C = πd
Area	A = πr²	A = C² ÷ (4π)
Radius	r = d ÷ 2	r = √(A ÷ π)

Example: r = 5 cm d = 2 × 5 = 10 cm C = 2 × π × 5 = 31.42 cm A = π × 5² = π × 25 = 78.54 cm²

Area grows with the square of the radius. Doubling the radius doubles the circumference but multiplies the area by four, which the radius chart in the results shows.

How to Find the Radius from Circumference or Area

From circumference: r = C ÷ (2π) C = 50 cm: r = 50 ÷ 6.2832 = 7.958 cm A = C² ÷ (4π) = 2,500 ÷ 12.566 = 198.94 cm²

From area: r = √(A ÷ π) A = 100 cm²: r = √(100 ÷ π) = √31.831 = 5.642 cm C = 2π × 5.642 = 35.45 cm

Measuring circumference with a tape is often easier than finding the exact center, for example around a tree trunk, a pipe, or a column. Divide by π to get the diameter directly: 50 ÷ π = 15.915 cm.

Arc Length, Sector Area, Chord, and Segment

An arc is part of the circumference. A sector is the wedge between two radii, like a slice of pizza. A chord is the straight line joining the ends of the arc, and the segment is the area between the chord and the arc.

θ in radians = degrees × π ÷ 180 Arc s = rθ · Sector = ½r²θ · Chord = 2r sin(θ ÷ 2) Segment = ½r²(θ − sin θ) · Height = r(1 − cos(θ ÷ 2))

Example: r = 10, θ = 60° = 1.0472 rad s = 10 × 1.0472 = 10.47; sector = ½ × 100 × 1.0472 = 52.36 chord = 2 × 10 × sin 30° = 10; segment = 50 × (1.0472 − 0.8660) = 9.06

Equation of a Circle: Standard and General Form

(x − h)² + (y − k)² = r² Center (2, −3), r = 4: (x − 2)² + (y + 3)² = 16 Expand: x² + y² − 4x + 6y + 4 + 9 − 16 = 0 General form: x² + y² − 4x + 6y − 3 = 0

General to standard. Complete the square: center = (−D ÷ 2, −E ÷ 2) and r = √((D ÷ 2)² + (E ÷ 2)² − F). With D = −4, E = 6, F = −3: center (2, −3) and r = √(4 + 9 + 3) = 4. If the value under the root is zero or negative, the equation is not a real circle.

Intercepts. Set y = 0 and solve for x: (x − 2)² = 16 − 9 = 7, so x = 2 ± 2.646. Setting x = 0 gives y = −3 ± 3.464.

Understanding Your Result: Units, π, and the Charts

Units. The unit menu labels the answers and does not convert them. A radius of 5 in gives a circumference of 31.42 in and an area of 78.54 in². To convert an area, square the length factor: 1 ft² = 144 in² because 1 ft = 12 in. Use the area calculator if you need the result in acres, hectares, or square meters.

Why π is never exact. π is irrational, so its decimal digits never end or repeat. The calculator uses the double precision value built into JavaScript, accurate to about 15 significant digits, which is far more precise than any physical measurement you enter.

Reading the charts. In Measurements mode, the solid line is area and the dashed line is circumference, plotted from a radius of 0 to twice your radius. The straight dashed line and the curved solid line show that circumference grows in proportion to r while area grows with r². In Arc mode, the doughnut shows your sector as a share of the whole circle. In Equation mode, the graph sets the x range from the chart width so both axes use nearly the same scale and the circle looks round.

Practical Examples and Common Mistakes

Pizza sizes. A 16 inch pizza has an area of 201.06 in², while a 12 inch pizza has 113.10 in². The larger one is only 4 inches wider but has (16 ÷ 12)² = 1.78 times the area.

Wheels. A 26 inch wheel covers its circumference, π × 26 = 81.68 inches, each turn, so it turns about 776 times per mile (63,360 inches).

Radius and diameter swapped. Putting the diameter into πr² makes the area four times too large.
Degrees in a radian formula. s = rθ needs θ in radians. Using 60 instead of 1.0472 gives an arc 57 times too long.
Square units. Circumference is a length (cm); area is in square units (cm²). A radius in feet gives area in square feet.
Rounding π early. Using 3.14 is about 0.05% low, and 22/7 is about 0.04% high. Fine for estimates, but keep full precision for engineering work.
Sign errors in equations. (x + 3)² means h = −3, not 3.

Frequently Asked Questions

How do you find the area of a circle?

Use A = πr², where r is the radius. A circle with a radius of 5 cm has an area of π × 25 = 78.54 cm². If you know the diameter, halve it first or use A = πd² ÷ 4. If you know the circumference, use A = C² ÷ (4π). The answer is always in square units.

How do you find the circumference of a circle?

Multiply the diameter by π, or twice the radius by π: C = πd = 2πr. A radius of 5 cm gives 2 × π × 5 = 31.42 cm. The circumference is the perimeter of the circle, so it uses ordinary length units, not square units. To measure a real object, wrap a flexible tape or a piece of string around it and measure the string.

How do you find the radius from the circumference?

Divide the circumference by 2π: r = C ÷ (2π). A 50 cm circumference gives r = 50 ÷ 6.2832 = 7.958 cm and a diameter of 15.915 cm. From the area, use r = √(A ÷ π), so an area of 100 cm² gives a radius of 5.642 cm. Choose "Circumference" or "Area" in the calculator to do this automatically.

How do you calculate arc length and sector area?

Convert the angle to radians, then use arc length s = rθ and sector area A = ½r²θ. For r = 10 and 60°, θ = 1.0472, so the arc is 10.47 and the sector is 52.36. In degrees, use (angle ÷ 360) × 2πr and (angle ÷ 360) × πr². Both methods give the same answer because a sector is a fraction of the whole circle.

What is the equation of a circle?

The standard form is (x − h)² + (y − k)² = r², with center (h, k) and radius r. Center (2, −3) and radius 4 give (x − 2)² + (y + 3)² = 16, which expands to the general form x² + y² − 4x + 6y − 3 = 0. To convert back, complete the square: the center is (−D ÷ 2, −E ÷ 2).

Sources & References

OpenStax. Prealgebra 2e, 9.5 Solve Geometry Applications: Circles and Irregular Figures (d = 2r, C = 2πr, A = πr²). openstax.org
OpenStax. Precalculus 2e, 5.1 Angles (radians, arc length s = rθ, sector area). openstax.org
Weisstein EW. Circle (circumference, area, center radius equation). Wolfram MathWorld. mathworld.wolfram.com
Weisstein EW. Circular Segment (chord, segment height, segment area). Wolfram MathWorld. mathworld.wolfram.com

Results are for educational purposes. Verify critical dimensions before ordering materials or making parts. Methodology reviewed: 2026.