Circles questions with answers

Questions

1. Question 1Circumference, in terms of π

   The diagram shows a circular garden pond. Its diameter is $36$ m. Find its circumference, leaving your answer in terms of $\pi$.

   Show the worked solution

   1. Step 1: The radius is half the diameter$r = \dfrac{36}{2}$$= 18$ m
   2. Step 2: Circumference = 2πr$C = 2\pi r$$= 2\pi(18)$$= 36\pi$ m
2. Question 2Area, in terms of π

   The diagram shows a circular fountain. Its radius is $21$ m. Find its area, leaving your answer in terms of $\pi$.

   Show the worked solution

   1. Step 1: Area = πr²$A = \pi r^2$$= \pi(21)^2$$= 441\pi$ m²
3. Question 3Circumference or area to 3 s.f.

   The diagram shows a circle with centre $O$ of radius $7$ cm. Using the $\pi$ key on your calculator, find the circumference of the circle, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Circumference = 2πr: use the π key, round at the end$C = 2\pi r$$= 2\pi(7)$$= 43.9822\ldots$$= 44.0$ cm (3 s.f.)
4. Question 4The radius from the circumference

   A circle has a circumference of $26\pi$ cm. Find its radius, $r$.

   Show the worked solution

   1. Step 1: Circumference = 2πr$2\pi r = 26\pi$
   2. Step 2: Divide both sides by 2π$r = \dfrac{26}{2}$$= 13$ cm
5. Question 5The radius from the area

   A circle has an area of $16\pi$ cm². Find its radius, $r$.

   Show the worked solution

   1. Step 1: Area = πr²$\pi r^2 = 16\pi$$r^2 = 16$
   2. Step 2: Take the positive square root$r = \sqrt{16}$$r = 4$ cm
6. Question 6Semicircles and quarter circles

   The diagram shows a semicircle of diameter $24$ m. Find its area, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: The radius is half the diameter$r = \dfrac{24}{2}$$= 12$ m
   2. Step 2: A semicircle is half of a circle$A = \dfrac{1}{2} \times \pi r^2$$= \dfrac{1}{2} \times \pi(12)^2$$= 226.1946\ldots$$= 226$ m² (3 s.f.)
7. Question 7Wheels: distance and turns

   The wheel of a trolley has a diameter of $42$ cm. Find the distance, in metres, it travels when it makes $40$ complete turns. Give your answer correct to 3 significant figures.

   Show the worked solution

   1. Step 1: One turn moves the wheel forward one circumference$\text{Circumference} = \pi d$$= \pi(42)$$= 131.9468\ldots$ cm
   2. Step 2: Multiply by the number of turns, then change cm to m$\text{Distance} = \dfrac{40 \times \pi(42)}{100}$$= 52.7787\ldots$$= 52.8$ m (3 s.f.)
8. Question 8A ring split into equal areas

   The diagram shows two circles with the same centre $O$, of radii $5$ m and $7$ m. The shaded ring between them is divided into $2$ parts of equal area by a dashed circle, also with centre $O$. Find the radius, $r$ m, of the dashed circle.

   Show the worked solution

   1. Step 1: Area of the shaded ring = big circle − small circle$\text{Ring} = \pi(7)^2 - \pi(5)^2$$= 24\pi$ m²
   2. Step 2: Each of the 2 parts is a half of the ring$\text{One part} = \dfrac{24\pi}{2}$$\phantom{\text{One part}} = 12\pi$ m²
   3. Step 3: Inside the dashed circle: the small circle and one part$\pi r^2 = \pi(5)^2 + 12\pi$$r^2 = 37$
   4. Step 4: Take the positive square root$r = \sqrt{37}$$= 6.0827\ldots$$= 6.08$ (3 s.f.)