Chapter 7 of 8

All 8 chapters
  1. 1Reflections and rotations, readunderstood
  2. 2Translations and combined moves, readunderstood
  3. 3Enlargements, readunderstood
  4. 4Congruent triangles, readunderstood
  5. 5Similar shapes, readunderstood
  6. 6Similar areas and volumes, readunderstood
  7. 7Arcs and sectors, readunderstood
  8. 8Angle rules in circles, readunderstood

M4.4.7 Chapter 7

Arcs and sectors

A quarter of a round flat bread has a radius of 10 cm. How long is its curved edge?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find the length of an arc of a circle
  • Find the area of a sector
  • Find the perimeter of a sector

A circle can be cut like a flat bread. A piece of the curved edge is an arc. A slice, with two straight edges and an arc, is a sector. The straight edges of a sector are both radii. They meet at the centre, O, and make an angle there.

A circle with centre O and points A and B on it. The radius OA is labelled radius. A shaded sector runs from A anticlockwise to B. The arc from A to B is drawn heavy and labelled arc. The angle at O is marked and labelled θ.radiusarcθABO
The shaded part is a sector. The heavy curve is its arc. The angle θ is at the centre.

A full turn is 360°. A sector with an angle of θ degrees is θ out of 360 of the whole circle. If θ = 90°, the sector is 90 out of 360, which is a quarter. If θ = 180°, it is a half.

Draw it first

A sector of 90° is a quarter of the circle

  1. A circle with centre O and points A and B on it. Step 1: the radii OA and OB are drawn, with the angle of 90° at O marked and labelled 90°. Step 2: the sector between the radii is shaded. Step 3: the arc from A to B is drawn heavy and labelled arc.90°ABO

    1 Draw two radii, OA and OB, with an angle of 90° between them.

  2. A circle with centre O and points A and B on it. Step 1: the radii OA and OB are drawn, with the angle of 90° at O marked and labelled 90°. Step 2: the sector between the radii is shaded. Step 3: the arc from A to B is drawn heavy and labelled arc.90°ABO

    2 Shade the sector between the radii. It is 90 out of 360 of the circle, which is a quarter.

  3. A circle with centre O and points A and B on it. Step 1: the radii OA and OB are drawn, with the angle of 90° at O marked and labelled 90°. Step 2: the sector between the radii is shaded. Step 3: the arc from A to B is drawn heavy and labelled arc.arc90°ABO

    3 The arc is the part of the circumference inside the sector. It is a quarter of the circumference.

This gives the rules. The circumference is 2πr and the area is πr². A sector takes the fraction θ ÷ 360 of each. Use the π key on your calculator, and say how you rounded.

Arc length = (θ ÷ 360) × 2πr. Sector area = (θ ÷ 360) × πr². Perimeter of a sector = arc length + 2r. The 2r is for the two straight edges.

A circle with centre O of radius 6 cm. The radius OA is labelled 6 cm. A shaded sector runs from A anticlockwise to B, with an angle of 60° at O labelled 60°. The arc from A to B is heavy and labelled arc.6 cmarc60°ABO
A sector with a radius of 6 cm and an angle of 60°.

Worked example

Find the length of the arc of a sector with a radius of 6 cm and an angle of 60°. Give the answer to 1 decimal place.

  1. To 1 decimal place, the whole circumference is 2 × π × 6 = 37.7 cm.
  2. The angle 60° is 60 out of 360. So the arc is 60 ÷ 360 of the circumference.
  3. To 1 decimal place, the arc length is 60 ÷ 360 × 37.7 = 6.3 cm.

Answer: 6.3 cm

A circle with centre O of radius 10 cm. The radius OA is labelled 10 cm. A shaded sector runs from A anticlockwise to B, with an angle of 72° at O labelled 72°. The arc from A to B is heavy and labelled arc.10 cmarc72°ABO
A sector with a radius of 10 cm and an angle of 72°.

Worked example

Find the area of a sector with a radius of 10 cm and an angle of 72°. Give the answer to 1 decimal place.

  1. To 1 decimal place, the area of the whole circle is π × 10 × 10 = 314.2 cm².
  2. The sector is 72 out of 360 of the circle. That is 72 ÷ 360 = 0.2.
  3. To 1 decimal place, the sector area is 0.2 × 314.2 = 62.8 cm².

Answer: 62.8 cm²

To walk all the way round the edge of a sector, you go along the arc and then along both radii. So the perimeter is longer than the arc.

A circle with centre O of radius 9 cm. The radius OA is labelled 9 cm. A shaded sector runs from A anticlockwise to B, with an angle of 120° at O labelled 120°. The arc from A to B is heavy and labelled arc.9 cmarc120°ABO
A sector with a radius of 9 cm and an angle of 120°.

Worked example

Find the perimeter of a sector with a radius of 9 cm and an angle of 120°. Give the answer to 1 decimal place.

  1. To 1 decimal place, the arc length is 120 ÷ 360 × 2 × π × 9 = 18.8 cm.
  2. The sector has two straight edges. Each is a radius of 9 cm.
  3. Together the straight edges are 2 × 9 = 18 cm.
  4. Perimeter = 18.8 + 18 = 36.8 cm.

Answer: 36.8 cm

See it three ways

With things

Cut a round piece of paper. Fold it in half, then in half again. Open it. Each fold line is a radius, and a quarter of the circle lies between two of them.

As a picture

A circle with centre O. The radius OA is labelled radius. A shaded sector of a quarter of the circle runs from A anticlockwise to B, with a right angle at O labelled 90°. The arc from A to B is heavy and labelled arc.radiusarc90°ABO

In symbols

Arc length = (θ ÷ 360) × 2πr. Sector area = (θ ÷ 360) × πr². Sector perimeter = arc length + 2r.

Try it

Measure an arc with string

You need: a round tin lid or a compass, string, a ruler, a pencil.

  1. Draw a circle with a radius of 5 cm. Mark its centre.
  2. Draw two radii at right angles to each other. Check the angle with the corner of a sheet of paper.
  3. Lay a piece of string along the arc between the radii. Straighten the string and measure it.

What you will see: The string is about 7.9 cm long. That matches 90 ÷ 360 × 2 × π × 5, which is about 7.85 cm.

If you cannot try it, this is what happens.
A circle with centre O of radius 5 cm. The radius OA is labelled 5 cm. A shaded sector runs from A anticlockwise to B, with an angle of 90° at O labelled 90°. The arc from A to B is heavy and labelled arc.5 cmarc90°ABO

A wrong idea many people have

The perimeter of a sector is the length of its arc.

Why it feels right: The arc is the curved edge, and the curved edge is the part that needs the new rule.

The test that breaks it: Trace round a slice of flat bread with a finger. You go along the curve, then along one straight edge, then along the other. A sector with a radius of 9 cm has 18 cm of straight edges.

What is true: The perimeter is the whole distance round the edge. It is the arc length plus two radii: arc + 2r.

Key words

arc
a part of the circumference of a circle.
sector
a slice of a circle between two radii. It has an arc for its curved edge.

Back to your guess

The answer is B: About 15.7 cm. That is a quarter of the distance round the circle.

The curved edge is a quarter of the circumference. The circumference is about 62.8 cm, because 2 × π × 10 is about 62.8. A quarter of that is about 15.7 cm. This chapter shows how to do the same for any slice.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A sector has a radius of 6 cm and an angle of 90°. How long is its arc, to 1 decimal place?

    A circle with centre O of radius 6 cm. The radius OA is labelled 6 cm. A shaded sector runs from A anticlockwise to B, with an angle of 90° at O labelled 90°. The arc from A to B is heavy and labelled arc.6 cmarc90°ABO
    Tap your answer.

    1. Not this one. You found the whole circumference. The arc is only a part of it, 90 out of 360. Go to M4.4.7. Try another.

    2. Yes, that is right. To 1 decimal place, the circumference is 2 × π × 6 = 37.7 cm. The sector is 90 out of 360, so the arc is 90 ÷ 360 × 37.7 = 9.4 cm.

    3. Not this one. You used the area rule, πr², and took 90 out of 360 of it. That is the area of the sector. An arc is a length, so use the circumference, 2πr. Go to M4.4.7. Try another.

    See the answer

    9.4 cm

    To 1 decimal place, the circumference is 2 × π × 6 = 37.7 cm. The sector is 90 out of 360, so the arc is 90 ÷ 360 × 37.7 = 9.4 cm.

    • If you chose A: You found the whole circumference. The arc is only a part of it, 90 out of 360. Go to M4.4.7.
    • If you chose C: You used the area rule, πr², and took 90 out of 360 of it. That is the area of the sector. An arc is a length, so use the circumference, 2πr. Go to M4.4.7.

  2. A sector has a radius of 9 cm and an angle of 40°. What is its area, to 1 decimal place?

    A circle with centre O of radius 9 cm. The radius OA is labelled 9 cm. A shaded sector runs from A anticlockwise to B, with an angle of 40° at O labelled 40°. The arc from A to B is heavy and labelled arc.9 cmarc40°ABO
    Tap your answer.

    1. Not this one. You found the area of the whole circle. The sector is only 40 out of 360 of it. Go to M4.4.7. Try another.

    2. Not this one. You used the arc length rule, with 2πr. An area uses πr², the area of the whole circle. Go to M4.4.7. Try another.

    3. Yes, that is right. The area of the whole circle is π × 9 × 9 = 254.5 cm². The sector is 40 out of 360, so its area is 40 ÷ 360 × 254.5 = 28.3 cm².

    See the answer

    28.3 cm²

    The area of the whole circle is π × 9 × 9 = 254.5 cm². The sector is 40 out of 360, so its area is 40 ÷ 360 × 254.5 = 28.3 cm².

    • If you chose A: You found the area of the whole circle. The sector is only 40 out of 360 of it. Go to M4.4.7.
    • If you chose B: You used the arc length rule, with 2πr. An area uses πr², the area of the whole circle. Go to M4.4.7.

  3. A sector has a radius of 5 cm and an angle of 90°. What is its perimeter, to 1 decimal place?

    Tap your answer.

    1. Yes, that is right. The arc is 90 ÷ 360 × 2 × π × 5 = 7.9 cm. The two radii are 2 × 5 = 10 cm. The perimeter is 7.9 + 10 = 17.9 cm.

    2. Not this one. You gave only the arc length. The perimeter goes all the way round, so add the two straight edges. Go to M4.4.7. Try another.

    3. Not this one. You added one radius. A sector has two straight edges, and both are radii. Add 2 × 5. Go to M4.4.7. Try another.

    See the answer

    17.9 cm

    The arc is 90 ÷ 360 × 2 × π × 5 = 7.9 cm. The two radii are 2 × 5 = 10 cm. The perimeter is 7.9 + 10 = 17.9 cm.

    • If you chose B: You gave only the arc length. The perimeter goes all the way round, so add the two straight edges. Go to M4.4.7.
    • If you chose C: You added one radius. A sector has two straight edges, and both are radii. Add 2 × 5. Go to M4.4.7.

  4. Which rule gives the length of an arc?

    Tap your answer.

    1. Not this one. Adding the angle to the radius mixes degrees and centimetres. An arc is a fraction of the circumference. Go to M4.4.7. Try another.

    2. Not this one. This does not give a length in centimetres. Degrees must first be turned into a fraction of 360. Go to M4.4.7. Try another.

    3. Not this one. This gives the area of a sector. An arc is a length, so it uses the circumference instead. Go to M4.4.7. Try another.

    4. Yes, that is right. An arc is a part of the circumference. The fraction of the circle is the angle ÷ 360, and the circumference is 2πr.

    See the answer

    Multiply the fraction of the circle by the circumference.

    An arc is a part of the circumference. The fraction of the circle is the angle ÷ 360, and the circumference is 2πr.

    • If you chose A: Adding the angle to the radius mixes degrees and centimetres. An arc is a fraction of the circumference. Go to M4.4.7.
    • If you chose B: This does not give a length in centimetres. Degrees must first be turned into a fraction of 360. Go to M4.4.7.
    • If you chose C: This gives the area of a sector. An arc is a length, so it uses the circumference instead. Go to M4.4.7.

  5. A sector has a radius of 4 m and an angle of 90°. What is its area, to 1 decimal place?

    Tap your answer.

    1. Not this one. You found the area of the whole circle. A sector with an angle of 90° is only a quarter of it. Go to M4.4.7. Try another.

    2. Yes, that is right. The whole circle has an area of π × 4 × 4 = 50.3 m². The sector is 90 ÷ 360 of it, which is 12.6 m².

    3. Not this one. The number is right, but the unit is not. An area is in square metres, m², and not in metres. Go to M4.4.7. Try another.

    See the answer

    12.6 m²

    The whole circle has an area of π × 4 × 4 = 50.3 m². The sector is 90 ÷ 360 of it, which is 12.6 m².

    • If you chose A: You found the area of the whole circle. A sector with an angle of 90° is only a quarter of it. Go to M4.4.7.
    • If you chose C: The number is right, but the unit is not. An area is in square metres, m², and not in metres. Go to M4.4.7.

  6. From an earlier chapter

    A shape has an area of 7 cm². It is enlarged by scale factor 3. What is the new area, in cm²?

    See the answer

    63

    The area scale factor is 3 × 3 = 9. The new area is 7 × 9 = 63 cm².

  7. A sector has a radius of 12 cm and an angle of 90°. How long is its arc, to 1 decimal place?

    See the answer

    18.8

    The circumference is 2 × π × 12 = 75.4 cm. The sector is 90 ÷ 360 of it, so the arc is 18.8 cm.

  8. A sector has a radius of 6 cm and an angle of 150°. What is its area, in cm², to 1 decimal place?

    See the answer

    47.1

    The area of the whole circle is π × 6 × 6 = 113.1 cm². The sector is 150 ÷ 360 of it, so its area is 47.1 cm².

  9. A sector has a radius of 8 cm and an angle of 45°. What is its perimeter, in cm, to 1 decimal place?

    See the answer

    22.3

    To 1 decimal place, the arc is 45 ÷ 360 × 2 × π × 8 = 6.3 cm. The two radii are 2 × 8 = 16 cm. The perimeter is 6.3 + 16 = 22.3 cm.

Keep it

Copy this into your exercise book.

Arc = (θ ÷ 360) × 2πr. Sector area = (θ ÷ 360) × πr². Sector perimeter = arc + 2r. Say how you rounded.

Draw: A circle with a shaded sector, its angle at the centre, the radius labelled, and the arc drawn heavy.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (1.0 MB). It opens on any phone.