Circumference Calculator
Calculate circumference, diameter, radius, and area of a circle from any one value. Includes arc length and sector area tools.
About Circumference Calculator
Calculate the circumference, radius, diameter, and area of any circle from a single measurement. The calculator also includes arc length and sector area tools for working with portions of a circle. Every result shows the formula and step-by-step substitution.
Circumference Formulas
The circumference (C) is the total distance around a circle:
C = 2πr = πd
And the reverse calculations:
| Known Value | Find Circumference | Find Radius |
|---|---|---|
| Radius (r) | C = 2πr | Given |
| Diameter (d) | C = πd | r = d/2 |
| Area (A) | C = 2π√(A/π) | r = √(A/π) |
| Circumference (C) | Given | r = C/(2π) |
Worked example: A tree trunk has a circumference of 157 cm (measured with a tape). Find its diameter:
- d = C / π = 157 / 3.14159 = 49.97 cm (about 50 cm diameter)
- r = 49.97 / 2 = 24.99 cm
- Cross-sectional area = πr² = π x 624.5 = 1,962 cm²
Arc Length and Sector Area
An arc is a portion of the circumference. A sector is the "pizza slice" area between two radii and the arc.
Arc length: L = (θ/360) x 2πr (degrees) or L = θr (radians)
Sector area: A = (θ/360) x πr² (degrees) or A = (1/2)θr² (radians)
Worked example: A 90° sector of a circle with radius 10 cm:
- Arc length = (90/360) x 2π(10) = (1/4) x 62.83 = 15.71 cm
- Sector area = (90/360) x π(100) = (1/4) x 314.16 = 78.54 cm²
- This is exactly 1/4 of the full circumference and area (as expected for a 90° angle)
Quick Reference: Circumferences of Common Objects
| Object | Diameter | Circumference |
|---|---|---|
| Golf ball | 42.67 mm | 134.0 mm |
| Tennis ball | 6.7 cm | 21.0 cm |
| Football (soccer) | 22 cm | 69.1 cm |
| Basketball | 24.3 cm | 76.3 cm |
| Car tyre (225/45R17) | 63.4 cm | 199.2 cm |
| Earth (equator) | 12,742 km | 40,075 km |
Why Circumference Matters
- Wheels and gears: Circumference determines how far a wheel travels per revolution. A 68 cm diameter bicycle wheel travels 213.6 cm per rotation, so 1 km = 468 revolutions.
- Pipes and tubing: When wrapping insulation or tape around a pipe, you need the circumference to know how much material one wrap requires.
- Running tracks: A standard 400m track is measured 30 cm from the inner edge. Each lane is ~1.22m wide, so each outer lane adds ~7.67m per lap (2π x 1.22).
- Trees: Foresters measure tree circumference (easier than diameter) and convert to diameter for timber volume calculations. Circumference in inches divided by π gives diameter.
Degrees vs Radians for Arc Calculations
| Degrees | Radians | Fraction of Circle |
|---|---|---|
| 360° | 2π (6.283) | Full circle |
| 180° | π (3.142) | Half circle |
| 90° | π/2 (1.571) | Quarter circle |
| 60° | π/3 (1.047) | Sixth of circle |
| 45° | π/4 (0.785) | Eighth of circle |
| 1° | π/180 (0.01745) | 1/360 of circle |
To convert: radians = degrees x π/180. To convert back: degrees = radians x 180/π.
For a complete visual reference of angles and trigonometric values, the unit circle shows sine, cosine, and tangent at every standard angle. For calculating the areas of other shapes, the area calculator covers rectangles, triangles, trapezoids, and more.
All calculations run in your browser. No data is sent anywhere.
Frequently Asked Questions
How do I calculate circumference from the radius?
Use the formula C = 2πr. Multiply the radius by 2 and then by pi (approximately 3.14159). For example, a radius of 5 gives a circumference of about 31.416.
Can I find the radius if I only know the area?
Yes. The formula is r = √(A/π). Enter the area and the tool will work backwards to give you the radius, diameter, and circumference.
What is arc length?
Arc length is the distance along a curved section of a circle. It depends on the central angle and the radius. The formula is L = (θ/360) x 2πr where θ is the angle in degrees.
What is the difference between circumference and perimeter?
For circles, circumference is the same thing as perimeter. It is the total distance around the outside of the circle.
How accurate are the results?
The tool uses JavaScript's built-in Math.PI which provides pi to about 15 decimal places. Results are shown up to 8 decimal places, which is more than enough for most practical and academic uses.
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