Find the perimeter of a sector (i.e., the curved portion as well as the two straight portions) of a circle of radius 14 cm and sector angle 75°.
Answer: Arc = 2 × 227 × 14 × 75360 = 553 ≈ 18.33 cm; perimeter = 18.33 + 14 + 14 = 1393 ≈ 46.33 cm.
Step-by-step solution
Given: Radius r = 14 cm; Sector angle θ = 75°
To find: Perimeter of the sector = arc + 2 radii
To find: Perimeter of the sector = arc + 2 radii
Idea: A sector’s boundary is the arc plus the two radii. Find the arc with 2πr × θ360° and add 2r.
- Arc length = 2πr × θ360° = 2 × 227 × 14 × 75360 = 88 × 75360.1 mark
- 88 × 75360 = 6600360 = 553 ≈ 18.33 cm.1 mark
- Perimeter = arc + 2r = 553 + 28 = 1393 ≈ 46.33 cm.1 mark
The perimeter of the sector is 139/3 cm ≈ 46.33 cm.
Check: 75° is a bit more than 15 of 360°, and 15 of 88 is 17.6, so an arc of 18.33 cm is reasonable ✓.
Answer to write in the exam
Arc = 2πr × θ360° = 2 × 227 × 14 × 75°360°
= 88 × 75360 = 553 ≈ 18.33 cm
Perimeter = arc + 2r = 18.33 + 28
∴ Perimeter ≈ 46.33 cm
Common mistakes that cost marks
- Giving only the arc (18.33 cm). The question asks for the two straight edges as well.
- Adding one radius instead of two.
- Using the area formula πr2 × θ360.
How this can come in the exam
MCQ (1 mark)
The perimeter of a sector of radius 7 cm and angle 90° is (use π = 227)
- 11 cm
- 18 cm
- 25 cm
- 38.5 cm
Show answer
(C) 25 cm
Arc = 14 × 44 = 11 cm; perimeter = 11 + 7 + 7 = 25 cm.
Short answer (2 marks)
A pizza slice is a sector of radius 21 cm with angle 60°. Find the length of crust around the slice (arc plus two straight cuts). (Use π = 227.)
Show answer
Arc = 2 × 227 × 21 × 60360 = 22 cm (1 mark). Perimeter = 22 + 21 + 21 = 64 cm (1 mark).Try one yourself
Find the perimeter of a sector of radius 10.5 cm and angle 120°. (Use π = 227.)
Show answer
Arc = 2 × 227 × 10.5 × 13 = 22 cm; perimeter = 22 + 21 = 43 cm.
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