Circles, sectors and arcs

Geometry and measures

Questions marked Higher are aimed at the Higher tier.

These are original practice questions, not a past paper. This site is independent and not affiliated with AQA or Pearson.

A circle has radius $7$ cm. Write its circumference in terms of $\pi$.

2 marks
A circle has radius $5$ cm. Write its area in terms of $\pi$.

2 marks
A circle has diameter $10$ cm. Using $\pi = 3.14$, find the circumference to $1$ decimal place.

3 marks
A sector has radius $6$ cm and angle $60°$. Find the arc length and the area in terms of $\pi$.

Sector of radius 6 cm and angle 60°

3 marks
A semicircle has radius $8$ cm. Find its area and its perimeter in terms of $\pi$.

4 marks
A sector has radius $9$ cm and angle $80°$. Find the arc length and the area in terms of $\pi$.

Sector of radius 9 cm and angle 80°

4 marks
A running track has two straight sides of $80$ m and two semicircular ends that together make a circle of radius $30$ m. Find the perimeter of the track and the area enclosed by it, in terms of $\pi$. Higher

5 marks
A slice of cake is a sector of radius $12$ cm and angle $45°$. Find the arc length and the area in terms of $\pi$. A ribbon runs along both straight edges and the arc. Find its length. Show that $8$ slices make a full circle. Higher

Cake slice: sector of radius 12 cm and angle 45°

6 marks

Mark scheme

Answer. $14\pi$ cm
- $C = 2\pi r$
- $14\pi$ cm
Answer. $25\pi$ cm$^{2}$
- $A = \pi r^{2}$
- $25\pi$ cm$^{2}$
Answer. $31.4$ cm
- $10 \times 3.14$
- $31.4$ cm
Answer. Arc $2\pi$ cm, area $6\pi$ cm$^{2}$
- The fraction of the circle is $\dfrac{60}{360} = \dfrac{1}{6}$.
- Arc $= \dfrac{1}{6} \times 12\pi = 2\pi$ cm.
- Area $= \dfrac{1}{6} \times 36\pi = 6\pi$ cm$^{2}$.
Answer. Area $32\pi$ cm$^{2}$, perimeter $8\pi + 16$ cm
- Area $= \dfrac{1}{2} \times \pi \times 64 = 32\pi$ cm$^{2}$.
- Curved length $= 8\pi$, and the diameter is $16$.
- Perimeter $= 8\pi + 16$ cm.
Answer. Arc $4\pi$ cm, area $18\pi$ cm$^{2}$
- Fraction $= \dfrac{80}{360} = \dfrac{2}{9}$.
- Arc $= \dfrac{2}{9} \times 18\pi = 4\pi$ cm.
- Area $= \dfrac{2}{9} \times 81\pi = 18\pi$ cm$^{2}$.
Answer. Perimeter $160 + 60\pi$ m, area $4800 + 900\pi$ m$^{2}$
- The two semicircles make a circumference of $2\pi \times 30 = 60\pi$.
- Perimeter $= 160 + 60\pi$ m.
- The straight section is an $80$ m by $60$ m rectangle, area $4800$.
- The circular ends have area $900\pi$, so the total area is $4800 + 900\pi$ m$^{2}$.
Answer. Arc $3\pi$ cm, area $18\pi$ cm$^{2}$, ribbon $24 + 3\pi$ cm
- Fraction $= \dfrac{1}{8}$.
- Arc $= 3\pi$ cm and area $= 18\pi$ cm$^{2}$.
- Ribbon $= 12 + 12 + 3\pi = 24 + 3\pi$ cm.
- $8 \times 18\pi = 144\pi = \pi \times 12^{2}$, the area of the full circle.