Sectors 鈥 area and perimeter
This module builds on: M1 Circumference and area of circles
A sector of a circle is an area enclosed by two
Sectors
A sector is a fraction of a circle. The angle in the sector is a fraction of 360掳.
This sector has an angle of 胃掳.
The fraction of the circle is \(\frac{胃}{360}\)
The area of the sector will be \(\frac{胃}{360}\) of the area \((蟺r^2)\) of the circle.
\(A = \frac{胃}{360} \times 蟺r^2\)
The arc length will be 胃/360 of the circumference (蟺d) of the circle.
Arc length \(= \frac{胃}{360} \times 蟺d\)
Example
- Calculate the area of the sector.
\(A = \frac{胃}{360} \times 蟺r^2\)
The area of the sector will be \(\frac{135}{360}\) of the area of the circle.
\(A = \frac{135}{360} \times 蟺 \times 4^2\)
\(A = 18.85 (2 d.p.)\)
- Calculate the perimeter of the sector.
The perimeter is made up of an arc and two straight sides.
Each straight side is a radius of the circle, r = 4
\(Arc~length = \frac{胃}{360} \times 蟺d\)
胃 = 135掳鈥 d = 2 x r = 8
\(Arc~length = \frac{135}{360} \times 蟺 \times 8 = 9.42 cm\)
Perimeter = arc + 2 radii
Perimeter = 9.42 + 2 x 4
= 17.42 cm (2d.p.)
Question
This fan that has 3 identical blades. Calculate the area of the blades.
Solution:
\(A = \frac{胃}{360} \times 蟺r^2\)
\(Area~of~each~blade = \frac{62}{360} \times 蟺 \times 28^2 = 424.1848鈥)
Answer: Total area = 3 x 424.1848 = 1272.55 cm虏 (2d.p.)
Question
Calculate the perimeter of this shape which is formed from two quarter circles.
Diameter = 16.4 cm
\(Arc~length = \frac{2}{4} \times 蟺 \times 16.4 = 25.76 cm\)
Each straight side is a radius of the circle, Radius = 8.2 cm
Perimeter = 25.76 + 8.2 + 8.2 = 42.16 cm (2 d.p.)
Answer: Perimeter = 42.16 cm (2 d.p.)
Test yourself
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