Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for geometry sector area and arc length? Then this Geometry lesson on Arc Length, Sector Area and Radian Measure explains each idea step by step, with worked examples.
Quick answer: Arc length and sector area are proportional to the central angle. An arc is the fraction θ/360 of the circumference 2πr, and a sector is the same fraction of the area πr². A radian is arc length divided by radius, so 180° equals π radians.
Standards: G-C.B.5 · Level: Grade 10 Geometry
How do you find the length of an arc?
An arc is a fraction of the circumference, and the fraction is the central angle over 360°. A sector is a fraction of the disk, with the same fraction.
s = θ360 · 2πr Area = θ360 · πr2
The two answers scale together: doubling the angle doubles both the arc and the area of the sector.
What is a radian?
The radian measure of a central angle is the length of its arc divided by the radius: θ = s ÷ r. A radian is the angle whose arc is exactly as long as the radius.
How do you convert between degrees and radians?
A full turn is 360°, which is 2π radians, so 180° = π radians. Multiply degrees by π/180, or radians by 180/π.
s = rθ Area = 12 r2θ
- Decide whether the angle is in degrees or radians.
- Write the fraction of the circle: θ/360 or θ/(2π).
- Multiply it by the circumference 2πr for an arc or by the area πr2 for a sector.
- Include units: linear units for arcs, square units for sectors.
A sector has radius 10 inches and a central angle of 72°. Find the arc length and the area.
The fraction is 72/360 = 15. Arc length = 15 · 20π = 4π ≈ 12.57 in. Area = 15 · 100π = 20π ≈ 62.83 in².
Using the circumference formula for a sector area, or the area formula for an arc. Arc length uses 2πr; sector area uses πr2.
- Arc length = (θ/360) · 2πr and sector area = (θ/360) · πr² with θ in degrees.
- Radian measure θ = s ÷ r; with radians, s = rθ and area = ½r²θ.
- 180° = π radians, so multiply degrees by π/180 to convert.
- Similar sectors with the same angle have arc lengths in the ratio of the radii.
Arc Length, Sector Area and Radian Measure: frequently asked questions
Why is arc length proportional to the central angle?
A circle has 360° around its center and one circumference around its edge. An angle that is a fraction of 360° cuts off the same fraction of the circumference.
What is one radian?
One radian is the central angle whose arc has the same length as the radius. Since the circumference is 2π radii long, a full circle measures 2π radians.
How do I convert 150° to radians?
Multiply by π/180: 150 · π/180 = 5π/6. The fraction 150/180 reduces to 5/6, so the angle is five sixths of π radians.
Why is the sector area ½r²θ in radians?
The sector is the fraction θ/(2π) of the disk. Multiplying by πr² leaves ½r²θ because the π cancels.
Do arc length and sector area use different units?
Yes. Arc length is a length in inches, feet or meters. Sector area is a two-dimensional measure in square inches, square feet or square meters.
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