A central angle has its vertex at the centre of a circle. It links arc length and radius and is the basis of the radian. Clock faces, pie charts, gear teeth and latitude differences on Earth are central angles.
How to use the Central Angle Calculator
- Known length
- Length
- Radius r
Results update as you type, and everything is calculated in your browser.
How it works
Choose whether you know the arc or the chord, then enter it with the radius. The arc gives θ = L/r directly; the chord uses the isosceles triangle formed with two radii.
Tips
- Angles are measured counter-clockwise from the positive x-axis by default.
- Add or subtract 360° (2π) to move between coterminal angles.
- Complementary angles add to 90°, supplementary angles to 180°.
- Use the unit menu to switch between degrees, radians and gradians.
Common mistakes to avoid
- Mixing up complementary and supplementary angles.
- Forgetting that a reference angle is always between 0° and 90°.
- Leaving an angle in radians when the question asks for degrees.
Inputs explained
| Field | What to enter |
|---|---|
| Known length | Choose one: Arc length L, Chord length c |
| Length | Units: cm, m, mm, in, ft, yd … |
| Radius r | Units: cm, m, mm, in, ft, yd … |
What each result means
- In radians = [In radians]([Central angle])
- Diameter = 2×[Radius r]
- Sector area = [Radius r]^2×[In radians]([Central angle])÷2
Example values
How the main result changes when Length changes and everything else stays at the example values above:
| Length (cm) | Central angle (°) |
|---|---|
| 5 | 35.8099 |
| 7.5 | 53.7148 |
| 10 | 71.6197 |
| 12.5 | 89.5247 |
| 15 | 107.4296 |
| 20 | 143.2394 |
Formula
θ = L / r (radians) = 180°·L/(πr)
From a chord: θ = 2·arcsin(c/(2r))
Worked example
Inputs — Known length: Arc length L; Length: 10 cm; Radius r: 8 cm.
Results — Central angle = 71.619724 °; In radians = 1.25 rad; Diameter = 16 cm; Sector area = 40 cm².
Key terms
- Central angle
- An angle at the centre of a circle.
Frequently asked questions
What is a central angle?
An angle whose vertex is the centre of the circle.
How is it related to the inscribed angle?
It is twice the inscribed angle on the same arc.
What is one radian?
The central angle whose arc equals the radius, ≈ 57.30°.
What is the central angle of a regular hexagon?
60°.