Sector area questions with answers

Questions

1. Question 1Sector area, angle in degrees

   A sector of a circle has radius $10$ cm and angle $72^\circ$. Find its area, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: The sector is a fraction of the circle$A = \dfrac{\theta}{360} \times \pi r^2$$\mathstrut A = \dfrac{72}{360} \times \pi(10)^2$
   2. Step 2: Work it out$A = 62.8318\ldots$$\phantom{A} = 62.8$ cm² (3 s.f.)
2. Question 2Sector area, angle in radians

   A sector of a circle has radius $12$ cm and angle $2.5$ radians. Find its area.

   Show the worked solution

   1. Step 1: In radians, area = ½r²θ$A = \dfrac{1}{2}r^2\theta$$\mathstrut A = \dfrac{1}{2} \times 12^2 \times 2.5$$A = 180$ cm²
3. Question 3Find the angle in radians

   A sector of a circle of radius $7$ cm has an area of $30$ cm². Find the angle, in radians, of the sector, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Substitute into A = ½r²θ, then make θ the subject$30 = \dfrac{1}{2} \times 7^2 \times \theta$$\theta = \dfrac{2 \times 30}{7^2}$$\theta = 1.2244\ldots$$\phantom{\theta} = 1.22$ rad (3 s.f.)
4. Question 4Find the radius

   A sector with angle $1.2$ radians has an area of $12$ cm². Find the radius, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Substitute into A = ½r²θ, then make r² the subject$12 = \dfrac{1}{2} \times r^2 \times 1.2$$r^2 = \dfrac{2 \times 12}{1.2}$
   2. Step 2: Square root$r = \sqrt{\dfrac{2 \times 12}{1.2}}$$r = 4.4721\ldots$$\phantom{r} = 4.47$ cm (3 s.f.)
5. Question 5Area from the arc length

   A sector of a circle of radius $10$ cm has an arc of length $5$ cm. Find the area of the sector.

   Show the worked solution

   1. Step 1: First the angle: θ = s ÷ r$\mathstrut \theta = \dfrac{5}{10}$$\theta = 0.5$
   2. Step 2: Then A = ½r²θ$\mathstrut A = \dfrac{1}{2} \times 10^2 \times 0.5$$A = 25$ cm²