# Central Angle Calculator

URL: https://speedycalculator.com/calculators/math/central-angle/
Last updated: 2026-10-30

> The central angle calculator finds the angle at the centre of a circle subtended by an arc, from the arc length and radius – or from the chord length and radius – in degrees and radians.

## Inputs

- Known length
- Length (cm, m, mm, in, ft, yd, km, mi)
- Radius r (cm, m, mm, in, ft, yd, km, mi)

## Results

- Central angle
- In radians
- Diameter
- Sector area

## Formula

```
θ = L / r (radians) = 180°·L/(πr)
From a chord: θ = 2·arcsin(c/(2r))
```

## Explanation

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

FieldWhat to enterKnown lengthChoose one: Arc length L, Chord length cLengthUnits: cm, m, mm, in, ft, yd …Radius rUnits: 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 (°)535.80997.553.71481071.619712.589.524715107.429620143.2394

## 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.

## FAQ

**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°.


## Sources

- [OpenStax – Prealgebra 2e](https://openstax.org/details/books/prealgebra-2e)
- [OpenStax – Algebra and Trigonometry 2e](https://openstax.org/details/books/algebra-and-trigonometry-2e)
- [Wolfram MathWorld – Central angle](https://mathworld.wolfram.com/CentralAngle.html)