Area of Sector

Worked example — sector area

Find the area of a sector with radius 10 cm and angle 108°.

Step by step

1. Substitute.

2. Evaluate.

Worked example — find the angle

A sector of radius 8 cm has area 32π cm².

Find its angle.

Step by step

1. Substitute into the area formula.

2. Cancel π and rearrange.

3. Solve for θ.

Worked example — find both arc and area

A sector has radius 6 cm and angle 150°.

Find:

(a) the arc length,

(b) the area.

Step by step

1. (a) Arc length.

2. (b) Sector area.

IB-style question — an annular sector (a ring)

A metal washer is shaped like an annular sector: a sector of a large circle with a smaller sector removed. The outer radius is 8 cm, the inner radius is 5 cm, and the central angle is 120°.

Find the area of the annular sector.

Step by step

1. Take the big sector and subtract the small sector — both share the 120° angle, so subtract the radii squared.

2. Substitute R = 8, r = 5, θ = 120°.

3. Evaluate.

Worked example — pizza slice

A circular pizza has diameter 30 cm.

It is cut into 8 equal slices.

Find the area of one slice.

Step by step

1. Each slice has angle θ = 360/8 = 45°. Radius = 15 cm (half of diameter).

2. Area of one slice.

IB-style question — a curved track (area → volume)

A curved section of running track is an annular sector with outer radius 30 m, inner radius 24 m and central angle 100°.

(a) Find the area of the track surface.

(b) The surface is covered with rubber 0.05 m deep. Find the volume of rubber needed.

Step by step

1. (a) Annular-sector area = (θ/360) × π(R² − r²) with R = 30, r = 24, θ = 100°.

2. Evaluate the area.

3. (b) The rubber is a thin layer: volume = surface area × depth.