Year 11 Standard · Unit 2 Right-angled triangles and circle geometry · Lesson 5 of 8

A slice of a circle is a fraction of the whole

A slice cut from a round tray takes a fraction of the circle's edge and of its area. This lesson turns that share into arc length, sector area and perimeter.

Geometry and TrigonometryGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 8: find the circumference of a circle of radius 7 cm, to one decimal place.
    Answer44.0 cm, because 2 × π × 7 ≈ 43.98.
  2. From Year 8: find the area of a circle of radius 6 cm, leaving π in the answer.
    Answer36π cm², because π × 6² = 36π.
  3. From Year 7: write 150/360 as a fraction in its simplest form.
    Answer5/12, because both numbers divide by 30.
Words used in this lesson
arc
a part of the curved edge of a circle. For example: the edge of a slice of pizza.
sector
a region between two radii and an arc. For example: a slice of a round cake.
circumference
the distance all the way round a circle. For example: the length of a ring of string.
central angle
the angle at the centre between two radii. For example: the point of the slice.
fraction of a circle
the part of 360° that an angle covers. For example: 90° out of 360° is a quarter.
perimeter
the length of the whole boundary of a shape. For example: the fence round a field.

Two slices of a round tray

A baker cuts two slices from round trays. One slice has an angle of 40° and comes from a tray of radius 18 cm. The other has an angle of 80° and comes from a tray of radius 9 cm. A strip of icing runs along the curved edge of each slice.

Predict first: which slice needs the longer strip of icing, or are they the same? Write down your guess and a reason, then use the first figure and the worked example to settle it.

150
5
13.09arc length
32.72sector area

Angle 150°, radius 5: that is 5/12 of the circle. Arc 5/12 × 2π × 5 ≈ 13.09, area 5/12 × π × 25 ≈ 32.72.

Show me the steps
  1. Angle 150°, radius 5: 150 ÷ 360 = 5/12 of the circle. Arc 5/12 × 2π × 5 ≈ 13.09, area 5/12 × π × 25 ≈ 32.72.
  2. Angle 60°, radius 6: 1/6 of the circle. Arc 1/6 × 2π × 6 = 2π ≈ 6.28.
  3. Angle 120° and radius 3: 1/3 of the circle. Arc 1/3 × 2π × 3 = 2π ≈ 6.28. Double the angle and half the radius give the same arc, but a smaller area.
  4. Angle 330°, radius 10: 11/12 of the circle. Arc ≈ 57.60 and area ≈ 287.98, nearly the whole circle, so the numbers are near 62.83 and 314.16.

A sector is the fraction t/360 of a circle, and both its arc and its area take that same fraction. Arc length = (t/360) × 2πr. Sector area = (t/360) × πr².

The perimeter of a sector is the arc plus the two straight radii: (t/360) × 2πr + 2r. To work back, treat t/360 as an unknown fraction and rearrange: t = 360 × arc ÷ 2πr, or t = 360 × area ÷ πr².

210
6
33.99perimeter
65.97area

Angle 210°, radius 6: arc ≈ 21.99, two radii 12, perimeter ≈ 33.99. The area, 65.97, is a different quantity in square units.

Show me the steps
  1. Angle 210°, radius 6: arc 7/12 × 2π × 6 ≈ 21.99. The two radii add 12, so the perimeter is ≈ 33.99.
  2. Angle 30°, radius 10: the arc is only 1/12 × 20π ≈ 5.24, but the two radii add 20, so the perimeter is ≈ 25.24. Straight edges can dominate.
  3. Angle 330°, radius 2: the arc is 11/12 × 4π ≈ 11.52, and the radii add 4. Now the arc is most of the perimeter.
  4. Whatever the angle, the perimeter is longer than the arc by exactly 2r. The area is measured in square units, so it cannot be added to a length.

Perimeter is a length around the edge, not just the curved part. A sector of radius 5 cm with an arc of 8 cm has a perimeter of 18 cm, because the two straight sides count too. And never add an area to a length: square centimetres and centimetres are different quantities.

Worked examples

1. The baker's slices. Find each arc length.

  1. First slice: 40/360 = 1/9, so arc = 1/9 × 2π × 18 = 4π ≈ 12.6 cm.
  2. Second slice: 80/360 = 2/9, so arc = 2/9 × 2π × 9 = 4π ≈ 12.6 cm. The two strips are the same length.

2. A sector has radius 9 cm and angle 100°. Find its arc length and its area, to 1 decimal place.

  1. 100/360 = 5/18. Arc = 5/18 × 2π × 9 = 5π ≈ 15.7 cm.
  2. Area = 5/18 × π × 81 = 22.5π ≈ 70.7 cm².

3. A sector has radius 12 cm and arc length 20 cm. Find its angle, to 1 decimal place.

  1. The circumference is 2π × 12 = 24π ≈ 75.40, so the fraction is 20 ÷ 75.40 ≈ 0.2653.
  2. Angle = 0.2653 × 360 ≈ 95.5°.

Your go. The first step is given. A sector has radius 5 cm and angle 252°. Find its area, in terms of π and then to 1 decimal place.

  1. 252 ÷ 360 = 7/10 of the circle.
  2. Multiply by the area of the whole circle, π × 5².
Check the answer

7/10 × 25π = 17.5π ≈ 55.0 cm². Check: the whole circle is about 78.5, and 7/10 of it is a little more than half, so about 55.

With a GDC (optional)

Use the π key, not 3.14, and enter the fraction of the circle with brackets: for example (150 ÷ 360) × 2 × π × 5. Round only at the end. To work back to an angle, rearrange by hand first so the unknown is on its own, then enter the whole expression once.

Check the size by hand: the arc must be less than the whole circumference, and the angle must be less than 360°. Everything here can be done with a scientific calculator, and an exact answer such as 4π is a good alternative to a decimal.

Check before you go on

1. A sector has radius 6 cm and angle 50°. Which expression gives the length of its curved edge?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A sector has radius 12 cm and angle 70°. How long is its arc, to 1 decimal place?

3. A sector has radius 15 cm and angle 48°. What is its area, to 1 decimal place?

Level 2 · Building

4. A sector of radius 8 cm has an arc length of 11 cm. What is the angle of the sector, to the nearest degree?

5. A sector has radius 7 cm and angle 135°. What is its perimeter, to 1 decimal place?

6. A sector of radius 8 cm has an area of 40 cm². What is its angle, to 1 decimal place?

Level 3 · Stretch

7. Two sectors share a centre and an angle of 75°. The larger has radius 12 cm and the smaller has radius 8 cm. What is the area of the band between them, to 1 decimal place?

8. A sector with an angle of 120° has an arc length of 25 cm. What is its radius, to 1 decimal place?

Something to think about

A sector's perimeter is exactly as long as the circumference of its whole circle. Is its angle more or less than 180°? Use the second figure to trap it between two settings, then find it exactly.

One way to think about it

More: about 245°. Set the perimeter equal to the circumference: (t/360) × 2πr + 2r = 2πr. The two radii must make up the missing arc, (1 − t/360) × 2πr = 2r, so 1 − t/360 = 1/π and t = 360 − 360/π ≈ 245.4°. The radius cancels, so every circle gives the same angle. In the figure, 240° falls just short and 270° goes past.

Exit ticket

9. A sector has radius 4 cm and angle 225°. What is its area, exactly?