The angle of a sector of a circle, radius 10.5cm, is 48°, Calculate the perimeter of the sector

Analyze the question We need the perimeter of a sector. The perimeter is made of the arc length plus the two radii.

Concept

• Arc length = (θ/360) × 2πr
• Perimeter of sector = arc length + 2r

Solution

Arc length = (48/360) × 2 × (22/7) × 10.5

• 2 × (22/7) × 10.5 = 2 × 22 × 1.5 = 66
• (48/360) × 66 = (2/15) × 66 = 132/15 = 8.8 cm

Two radii = 2 × 10.5 = 21 cm

Perimeter = 8.8 + 21 = 29.8 cm

Check against the options

• A (8.8 cm) is only the arc length, so it leaves out the two radii.
• D (29.8 cm) is the arc length plus the two radii, which is the full perimeter.

Tip "Perimeter of a sector" always means arc + 2 radii. Examiners often put the arc length alone as an option to catch students who forget the radii. Also, when you see a radius like 10.5 or 21, use π = 22/7, since the numbers cancel neatly.