Sector Area
The area of a sector depends on the radius and the angle at the centre.
Key idea
For a circle of radius r and a central angle θ measured in radians, the area of the sector is:
A = ½ r²θ
If the angle is given in degrees, first convert to radians: θ (radians) = θ (degrees) × π / 180.
Interactive Sector Area Lab
Drag the red point, or use the sliders, to explore how sector area changes.
Drag the red point on the circle. When close to special angles, it gently snaps (π/6, π/4, π/3, π/2, etc.). A right-angle marker appears at 90°.
Radius:
Angle: °
Angle in radians:
Sector area A = ½ r²θ:
Working (radians):
θ (radians) = θ° × π / 180
θ = × π / 180 =
A = ½ r²θ = ½ × ² × =
Working (degrees):
A = (θ° / 360) × πr²
A = ( / 360) × π × ²
=
Find the area of a sector
A circle has radius 6 cm. Find the area of the sector that subtends an angle of 120° at the centre.
Given:
r = 6 cm
θ = 120°
Step-by-step
- Convert the angle to radians:
θ = 120° × π / 180 = (2π/3) radians. - Use the formula A = ½ r²θ:
A = ½ × 6² × (2π/3) = 12π cm². - Approximate if needed:
A ≈ 12π ≈ 37.70 cm² (to 2 d.p.).
The sector area is 12π cm² (≈ 37.70 cm²).
Try it yourself
Click “New question” to generate a random radius and angle. Work it out, then reveal the answer.
Radius: units
Angle: °
© Alexander Forrest