Arc lengths & sectors · Ages 14–15
Perimeter of a sector
Perimeter = arc + 2r.
How to do it
Method
Perimeter = arc + 2r
r = 6, θ = 60°: 2π + 12 cm
Another way
Walk right round
6 + 2π + 6
In full
A sector's perimeter is the distance right round it: one arc and two radii. Perimeter = arc + 2r. Radius 6 cm, angle 60°: the arc is 2π cm, so the perimeter is 2π + 12 cm; with π as 3.14, 6.28 + 12 = 18.28 cm. The commonest slip is the arc alone, the two radii left out. Or walk round it: along one radius out to the circle (6), along the arc (2π), and back down the other radius (6). A semicircle's perimeter is πr + 2r: its arc and a diameter. The other way round, from the perimeter and the radius: arc = perimeter − 2r, and then the angle.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The sector in the picture has a radius of 10 cm and an angle of 90°. Its arc is π cm, and its perimeter (π + ) cm.
Fill in each blank
Show the answer and working
Answer: 5, 5, 20
Arc 90360 × 2π × 10 = 5π; the perimeter adds two radii, 2 × 10 = 20.
Example 2
Taking π as 3.14, a sector of radius 18 cm and angle 180° has a perimeter of 92.52 cm.
True or false?
Show the answer and working
Answer: True
Perimeter = arc + two radii = 92.52 cm; the arc alone leaves out 2 × 18.
Example 3
Taking π as 3.14, a sector with an angle of 45° has a perimeter of 30.635 cm. What is its radius, in cm?
Show the answer and working
Answer: 11 cm
Perimeter = 45360 × 2 × 3.14 × r + 2r = 2.785r, so r = 30.635 ÷ 2.785 = 11 cm.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The sector in the picture has a radius of 5 cm and an angle of 180°. Its arc is π cm, and its perimeter (π + ) cm.
Fill in each blank
Answer:
2.The sector in the picture has a radius of 15 cm and an angle of 120°. What is its perimeter?
- A. (10π + 15) cm
- B. 10π cm
- C. (10π + 30) cm
- D. (10π × 30) cm
Choose the right answer
Answer:
3.Taking π as 3.14, a sector of radius 3 cm and angle 120° has a perimeter of 12.28 cm.
True or false?
Answer:
4.The sector in the picture has a radius of 22 cm and an angle of 70°. Correct to 1 decimal place, what is its perimeter, in cm?
Answer:
5.Taking π as 3.14, a sector with an angle of 180° has a perimeter of 77.1 cm. What is its radius, in cm?
Answer:
All the answers
- 1. 5, 5, 10
- 2. C. (10π + 30) cm
- 3. True
- 4. 70.9 cm
- 5. 15 cm
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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