Central Angle Calculator
Enter any two of the radius, arc length and central angle to find the third.
Updated
Central Angle Calculator
Our Central Angle Calculator links the three measurements of a circular arc: the central angle, the arc length and the radius. Give it any two and it finds the third, in degrees or radians. The working, the exact answer, the diameter, the circumference, the sector area and a drawing are shown as you type.
Quick answer
The central angle in radians is arc length ÷ radius: θ = 15 ÷ 10 = 1.5 rad ≈ 85.94°.
- In degrees: θ° = (L ÷ r) × 180 ÷ π
- Arc length: L = r × θ (θ in radians)
- Radius: r = L ÷ θ (θ in radians)
- One radian: the arc equals the radius, about 57.3°
What Is a Central Angle?
A central angle (θ) is the angle at the centre of a circle between two radii. The two radii meet the circle at the ends of an arc, and the length of that arc (L) grows with both the angle and the radius: a wider angle or a bigger circle both give a longer arc.
Central Angle Formula
With the angle in radians, the three values are tied together by one simple rule, L = rθ. Rearrange it for whichever value is missing.
| You want | Formula |
|---|---|
| Angle (radians) | θ = L ÷ r |
| Angle (degrees) | θ = (L ÷ r) × 180 ÷ π |
| Arc length | L = r × θ |
| Radius | r = L ÷ θ |
With the angle in degrees, use L = (θ ÷ 360) × 2πr and r = 180L ÷ (πθ). For example, a radius of 10 and an arc of 15:
θ = 15 ÷ 10 = 1.5 rad
θ = 1.5 × 180 ÷ π
θ ≈ 85.9437°
What Is a Radian?
One radian is the central angle whose arc is exactly as long as the radius. That is why θ = L ÷ r works so neatly: it counts how many radii fit along the arc. A full circle is 2π radians (about 6.28 radii of arc), so 1 radian is 360° ÷ 2π ≈ 57.2958°.
Central Angle Examples
| You know | Answer |
|---|---|
| r = 10, L = 15 | θ ≈ 85.94° |
| r = 8, L = 8 | θ = 1 rad ≈ 57.30° |
| r = 5, θ = 60° | L = 5π/3 ≈ 5.24 |
| r = 6, θ = 2 rad | L = 12 |
| L = 20, θ = 90° | r = 40/π ≈ 12.73 |
Angle as a Share of the Circle
The central angle is the fraction of the full turn that the arc covers. A 90° angle is a quarter of 360°, so its arc is a quarter of the circumference: with a radius of 10, that is 2π × 10 ÷ 4 ≈ 15.71. The same share applies to the sector area.
- Pie charts: a slice showing 25% has a central angle of 90°.
- Clock faces: each hour mark is 30° apart, each minute 6°.
- Wheels and gears: a wheel of radius 30 cm turning 1 radian rolls 30 cm.
Common Mistakes
- Degrees in L = rθ: the formula needs radians. 10 × 60 is 600, not the 10.47 you get from 10 × π/3.
- Diameter for radius: θ = L ÷ r uses the radius, half the diameter.
- Mixed units: the arc and the radius must be in the same unit before dividing.
- Arc longer than the circle: L can be at most 2πr, which is a full 360°.
Key Terms
- Central angle
- The angle at the centre between two radii, θ.
- Arc length
- The distance along the circle between the radii, L.
- Radius
- From the centre to the edge, r.
- Radian
- The angle whose arc equals the radius, about 57.3°.
How to Use the Central Angle Calculator
- Choose what to solve for. Pick Angle, Arc length or Radius. The two fields you need appear below.
- Enter the two known values. Type the numbers, such as a radius of 10 and an arc length of 15. Pick a length unit if you want one, and deg or rad for the angle. Decimals, fractions such as 3/4 and angles such as π/3 work.
- Read and copy the result. The answer appears as you type, with the working, the exact form, the angle in the other unit, the diameter, the circumference, the share of the circle, the sector area and a drawing. Press Copy to copy it, or Clear or Esc to start again.
Example: a curved garden path follows a circle with a radius of 10 m and is 15 m long. Choose Angle, enter 10 and 15 and pick m. The path turns through about 85.94°, or 1.5 rad.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
How do you find the central angle of a circle?
Divide the arc length by the radius: θ = L ÷ r gives the angle in radians. Multiply by 180 ÷ π for degrees. A radius of 10 and an arc of 15 give 1.5 rad, about 85.94°.
How do you find the arc length from the central angle?
Multiply the radius by the angle in radians: L = rθ. In degrees, use L = (θ ÷ 360) × 2πr. A radius of 5 and an angle of 60° give 5π/3 ≈ 5.24.
How do you find the radius from the arc length and central angle?
Divide the arc length by the angle in radians: r = L ÷ θ. An arc of 20 with a 90° angle (π/2 rad) gives r = 40/π ≈ 12.73.
What is the sum of the central angles of a circle?
360°, or 2π radians. Central angles that fill the whole circle without overlapping, such as the slices of a pie chart, always add up to a full turn.
Is a central angle the same as an inscribed angle?
No. A central angle has its vertex at the centre of the circle. An inscribed angle has its vertex on the circle and is half the central angle that cuts off the same arc.
Is the Central Angle Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Central Angle Calculator to solve for any circle's angle, arc length or radius instantly.