Teach Me Maths

Circles and Semicircles

On this page

What arcs, sectors and segments are, and how to work out the curved edge, the area and the perimeter of quarter circles and semicircles, with pi left in or rounded, including in bigger shapes such as garden plans.

Circles and semicircles in the exam

An arc is part of the edge of a circle. A sector is a slice of a circle, like a slice of pizza, with two radii as its straight edges. On the Foundation papers the sectors are quarter circles and semicircles.

You may be asked for the area of a quarter circle or a semicircle, the length of its curved edge, or the distance all the way round it. Harder questions, for 4 or 5 marks, put one into a bigger shape such as the plan of a garden or a field, ask for the area of a segment, or end with a cost or a number of bags to buy.

Paper 1 has no calculator, so there you keep \pi as it is and write an area such as 25\pi. On Papers 2 and 3 you use the \pi key and round, for example correct to 3 significant figures.

Since 2022 a formulae sheet has been given in the exam, with the circumference and area of a circle on it, but learn them anyway. Quarter circles and semicircles come from those formulae, as a fraction of the whole circle.

Semicircles and quarter circles

A semicircle is half a circle. Its straight edge is a diameter. A quarter circle is a quarter of a circle. Its two straight edges are radii, at right angles to each other.

The curved edge of each shape is an arc: part of the circumference. A question may call a quarter circle a sector of a circle with an angle of 90^\circ at the centre.

Points spaced equally round a circle cut the circumference into equal arcs. With 5 points, each arc is \frac{1}{5} of the circumference. For a circle with diameter 10 m, each arc is \pi\times10\div5=6.28\ldots m.

Curved edges and areas

Work out the whole circle first, then divide: by 2 for a semicircle, by 4 for a quarter circle. For the whole circle,

\text{circumference}=\pi d

\text{area}=\pi r^2

where d is the diameter and r is the radius. Only the radius is squared.

The curved edge is that fraction of the circumference. The area is the same fraction of the area of the circle.

For example, a quarter circle with radius 10 cm has arc length \frac{1}{4}\times\pi\times20=5\pi cm and area \frac{1}{4}\times\pi\times10^2=25\pi\text{ cm}^2

For an answer in terms of \pi, work out the number and write \pi after it, as in 25\pi. For a rounded answer, use the \pi key on your calculator and round only at the end.

Worked example 1: Area of a quarter circle without a calculator

Do not use a calculator. A mirror is a quarter circle with radius 14 cm. Work out the area of the mirror. Give your answer in terms of \pi

  1. The mirror is \frac{1}{4} of a whole circle with radius 14 cm.
  2. Whole circle: \pi\times14^2=196\pi\text{ cm}^2. Keep \pi as it is.

  3. Your turn. The mirror is one quarter of the whole circle. What number goes in front of \pi in the area of the mirror?

    196\div4=49, so the mirror is 49\pi\text{ cm}^2

  4. Answer: 49\pi\text{ cm}^2

Worked example 2: The curved edge of a semicircle

A garden arch is a semicircle with diameter 1.6 m. A string of lights goes along the curved edge of the arch. Work out the length of the curved edge. Give your answer correct to 1 decimal place.

  1. The curved edge is half the circumference of a circle with diameter 1.6 m.
  2. Whole circumference: \pi\times1.6=5.026\ldots m

  3. Your turn. Halve 5.026\ldots and give your answer correct to 1 decimal place.

    5.026\ldots\div2=2.513\ldots, which is 2.5 to 1 decimal place.

  4. Check: \pi is a bit more than 3, so the curved edge is a bit more than 3\times1.6\div2=2.4 m. So 2.5 m is sensible.
  5. Answer: 2.5 m

Perimeter of a quarter circle or a semicircle

The perimeter is the distance all the way round a shape. A quarter circle has three edges: the arc and two radii. So

\text{perimeter}=\text{arc length}+r+r

where r is the radius.

The quarter circle with radius 10 cm has arc length 5\pi=15.707\ldots cm. Its perimeter is 15.707\ldots+10+10=35.707\ldots, which is 35.7 cm to 1 decimal place.

For a semicircle the two radii make one straight line, the diameter. Its perimeter is the curved edge plus the diameter.

For a shape made from a rectangle and a semicircle, add only the edges round the outside. A line where the two parts meet is inside the shape, so it is not part of the perimeter.

Worked example 3: Perimeter of a quarter circle

A pond is a quarter circle with radius 1.8 m. A low fence goes all the way round the pond. Work out the length of the fence. Give your answer correct to 1 decimal place.

  1. All the way round means the curved edge and the two straight edges, which are radii.
  2. The diameter is 2\times1.8=3.6 m, so the whole circumference is \pi\times3.6=11.309\ldots m
  3. Curved edge: 11.309\ldots\div4=2.827\ldots m

  4. Your turn. Add the two radii. Work out 2.827\ldots+1.8+1.8, correct to 1 decimal place.

    2.827\ldots+1.8+1.8=6.427\ldots, which is 6.4 to 1 decimal place.

  5. Answer: 6.4 m

Worked example 4: A rectangle with a semicircle on top

A window is a rectangle 90 cm wide and 120 cm tall, with a semicircle on top. The diameter of the semicircle is the top of the rectangle. A metal strip goes all the way round the outside of the window. Work out the length of the metal strip. Give your answer correct to the nearest centimetre.

  1. The top of the rectangle is inside the window, so it is not part of the way round.

  2. Your turn. Work out the total length of the straight edges round the outside of the window.

    120+90+120=330 cm

  3. Curved edge: half of \pi\times90, which is 282.743\ldots\div2=141.371\ldots cm
  4. Total: 330+141.371\ldots=471.371\ldots cm, which is 471 cm to the nearest centimetre.
  5. Answer: 471 cm

Segments and longer problems

A segment is the part of a circle between a chord and the arc. It is not the same as a sector: a sector goes in to the centre, but a segment does not.

For a quarter circle, the segment is the sector with a right-angled triangle taken away:

\text{segment area}=\text{sector area}-\text{triangle area}

The triangle has two radii as its base and height, so its area is \frac{1}{2}\times r\times r

In a longer question, split the shape into parts such as rectangles, triangles, quarter circles and semicircles. Work out each area, then add or subtract. Keep the full calculator value until the end, and round a number of bags or packets up.

You may be asked to compare two shapes. A quarter circle with radius 12 cm has area \frac{1}{4}\times\pi\times12^2=36\pi\text{ cm}^2. A whole circle with radius 3 cm has area \pi\times3^2=9\pi\text{ cm}^2. As 36\div9=4, the area of the quarter circle is 4 times the area of the circle.

Worked example 5: Gravel for a patio

A patio is a rectangle 8 m long and 3 m wide, with a quarter circle of radius 3 m on one end, as in the diagram. The patio is going to be covered with gravel. One bag of gravel covers 4\text{ m}^2. Each bag costs £4.85. Work out the total cost of the bags of gravel needed.

  1. Split the patio into the rectangle and the quarter circle.
  2. Rectangle: 8\times3=24\text{ m}^2

  3. Your turn. Work out the area of the quarter circle, correct to 2 decimal places.

    \pi\times3^2\div4=7.068\ldots, which is 7.07\text{ m}^2 to 2 decimal places.

  4. Total area: 24+7.068\ldots=31.068\ldots\text{ m}^2
  5. Bags: 31.068\ldots\div4=7.76\ldots. So 7 bags are not quite enough: round up to 8 bags.
  6. Answer: 8\times\pounds 4.85=\pounds 38.80

Worked example 6: Seeds for a segment

The diagram shows a quarter circle OPQ with centre O and radius 6 m. The shaded segment, between the chord PQ and the arc, is a flower bed. One packet of seeds covers 2.5\text{ m}^2 of ground. Work out how many packets of seeds are needed to cover the flower bed.

  1. The flower bed is the quarter circle with triangle OPQ taken away.
  2. Quarter circle: \pi\times6^2\div4=9\pi=28.274\ldots\text{ m}^2
  3. Triangle OPQ has a right angle at O: \frac{1}{2}\times6\times6=18\text{ m}^2

  4. Your turn. Work out the area of the flower bed, 28.274\ldots-18, correct to 2 decimal places.

    28.274\ldots-18=10.274\ldots\text{ m}^2, which is 10.27 to 2 decimal places.

  5. Packets: 10.274\ldots\div2.5=4.10\ldots. So 4 packets are not quite enough: round up.
  6. Answer: 5 packets

Common mistakes

  • Forgetting to divide by 4 for a quarter circle. Work out the whole circle, then divide by 4, or by 2 for a semicircle.
  • Using the diameter as the radius. Halve the diameter before you put it into \pi r^2
  • Squaring the wrong thing. Only the radius is squared: \pi\times8^2=64\pi, so do not square \pi or forget to square 8
  • Finding the circumference when you need the area. An area uses \pi r^2 and is in square units; a curved edge comes from \pi d
  • Not rounding the number of bags up. If you need 7.76 bags, buy 8, because 7 bags are not enough.
  • Mixing up a sector and a segment. A sector goes in to the centre; a segment is the part between a chord and the arc.

Practice questions

More on this topic: Circles and Semicircles worksheet with full solutions