How do you find a central angle from arc length?

Use θ = s/r, where s is the arc length and r is the radius. The result is in radians. Convert to degrees by multiplying by 180/π if needed.

How do you find a central angle from sector area?

Start with A = 1/2 r²θ and solve for θ. The result is θ = 2A/r². This works when θ is in radians.

How do you find a central angle from chord length?

The chord relationship is chord = 2r sin(θ/2). Solving for θ gives θ = 2 sin⁻¹(chord/(2r)). The chord cannot be longer than the diameter.

Is a central angle the same as an inscribed angle?

No. A central angle has its vertex at the center of the circle. An inscribed angle has its vertex on the circle. When they intercept the same arc, the inscribed angle is half the central angle.

Education

Central Angle Calculator

Find a circle central angle from arc length, sector area, or chord length, with radians, degrees, and formula steps. This page is written for students who need both a fast answer and a clear explanation of the method, so it includes the formula, steps, examples, common mistakes, and practical notes.

What a central angle is

A central angle has its vertex at the center of a circle. Many circle formulas are really central-angle formulas in disguise: arc length uses θ = s/r, sector area uses θ = 2A/r², and chord problems use a sine relationship. The OpenStax angles lesson explains why radian measure naturally connects a central angle to arc length.

Ways to find a central angle

From arc length: θ = s ÷ r From sector area: θ = 2A ÷ r² From chord length: θ = 2 sin⁻¹(chord ÷ 2r)

Radians are the most direct unit for the calculation, but the calculator also shows degrees because most geometry problems ask for angle measures in degrees.

Common questions

Use θ = s/r, where s is the arc length and r is the radius. The result is in radians. Convert to degrees by multiplying by 180/π if needed.
Start with A = 1/2 r²θ and solve for θ. The result is θ = 2A/r². This works when θ is in radians.
The chord relationship is chord = 2r sin(θ/2). Solving for θ gives θ = 2 sin⁻¹(chord/(2r)). The chord cannot be longer than the diameter.
No. A central angle has its vertex at the center of the circle. An inscribed angle has its vertex on the circle. When they intercept the same arc, the inscribed angle is half the central angle.