How to use the circle calculator
- Choose what you know: the radius, diameter, circumference or area, and enter that one value.
- Read the radius, diameter, circumference and area. Each is derived from the one you entered.
- Open the advanced options and enter a sector angle in degrees (0 to 360) to also get the arc length, sector area, chord length and segment area.
- Use any consistent unit for lengths. Areas are in the square of that unit.
Formula
Diameter d = 2r, circumference C = 2 pi r, area A = pi r^2
From circumference: r = C / (2 pi). From area: r = sqrt(A / pi)
Arc length = r x angle (radians). Sector area = r^2 x angle / 2
Chord = 2 r sin(angle / 2). Segment area = r^2 (angle - sin(angle)) / 2
The angle in the sector formulas is the central angle in radians, which is degrees x pi / 180. Pi is the double-precision value, and results are shown to 8 decimal places or fewer.
Worked example
A circle with radius 5 has a diameter of 10, a circumference of 2 x pi x 5 = 31.41592654 and an area of pi x 25 = 78.53981634.
If you know only that the area is 100, the radius is sqrt(100 / pi) = 5.64189584. If the circumference is 2 pi, the radius is exactly 1.
For a radius of 2 and a 90 degree angle, the arc length is 3.14159265, the sector area is 3.14159265, the chord is 2.82842712 and the segment area is 1.14159265.
Relationships to remember
Everything about a circle follows from its radius. Doubling the radius doubles the diameter and circumference but multiplies the area by four. The ratio of circumference to diameter is always pi, about 3.14159, which is how pi is defined. For a calculation involving the surface of a flat shape, see the area calculator, and for round solids such as cylinders and spheres use the volume calculator.
Arcs, sectors and segments
A sector is a pie slice bounded by two radii and an arc. A segment is the region between a chord and its arc, like the cap cut off by a straight line. For a central angle theta in radians, the arc is r times theta, the sector is half of r squared times theta, and the segment is the sector minus the triangle formed by the two radii and the chord.
Common mistakes
A few errors come up repeatedly.
- Using the diameter in a formula that needs the radius. Divide by 2 first, or choose diameter in the dropdown.
- Mixing units, such as a radius in inches and an area in square feet.
- Entering the angle in radians when the field expects degrees.
- Rounding pi to 3.14 early, which changes the fourth significant digit of results.
Assumptions and limits
- The shape is a perfect circle in a flat plane. Real circular objects may be slightly out of round.
- The value must be greater than 0. Zero and negative values are rejected.
- Sector angles must be between 0 and 360 degrees. The segment formula also works for angles above 180 degrees.
- Results are shown to 8 decimal places or fewer, rounded half up.
- Everything runs in your browser. Nothing you enter is sent anywhere.
Frequently asked questions
How do you find the area of a circle?
Multiply pi by the radius squared. A circle with radius 5 has an area of about 78.54 square units. If you know the diameter, halve it first.
How do I find the radius from the circumference?
Divide the circumference by 2 pi. A circumference of 31.41592654 gives a radius of 5.
What is the difference between a sector and a segment?
A sector is a pie-shaped slice between two radii. A segment is the area between a chord and its arc. The segment equals the sector minus the triangle.
What value of pi does this calculator use?
It uses the double-precision value of pi, about 3.141592653589793, which is accurate far beyond what the 8 shown decimals need.
How do I find the length of a chord?
Use 2 times the radius times the sine of half the central angle. For a radius of 2 and a 90 degree angle, the chord is 2.82842712.