Year 11 Extended · Unit 1 Right-angled triangles and circle geometry · Lesson 4 of 7

Arcs, sectors and segments: a fraction of a circle, and how it scales

A sector is a fraction of a circle, so its arc and its area are the same fraction of the circle's circumference and area. This lesson uses that fraction forwards and backwards, finds exact answers in terms of π, and takes a triangle away to find a segment.

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: the circumference of a circle of radius 5 cm, in terms of π.
    Answer10π cm.
  2. From Year 8: the area of a circle of radius 6 cm, to 1 decimal place.
    Answer113.1 cm², from π × 36.
  3. From Year 9: a triangle has two sides of 8 cm and 8 cm with a 90° angle between them. Find its area.
    Answer32 cm², from ½ × 8 × 8.
Words used in this lesson
arc
a part of the circumference of a circle. For example: the crust of a pizza slice.
sector
the region between two radii and an arc. For example: a slice of cake.
segment
the region between a chord and the arc it cuts off. For example: the cap cut off a circle by a straight line.
chord
a straight line joining two points on a circle. For example: a bowstring.
perimeter
the total length around the outside of a shape. For example: the fence around a garden.

Slicing a round cake

A baker cuts a sector from a round cake of radius 6 cm with a 120° angle at the centre. A second cake of radius 12 cm is cut with the same 120° angle.

Predict first: the second cake has twice the radius. Does the second slice have twice the crust length (arc)? Does it have twice the top area? Write what you think, then test it with the first figure.

6
120
12.6 cmarc length
37.7 cm²area of the sector

Radius 6, angle 120°: one third of the circle, arc 12.6 cm, area 37.7 cm², perimeter 24.6 cm.

Show me the steps
  1. Radius 6, angle 120°: the fraction is 120 ÷ 360 = 1/3. The arc is 1/3 of 2π × 6 ≈ 12.6 cm and the area is 1/3 of π × 6² ≈ 37.7 cm².
  2. Double the radius to 12 and keep 120°: the arc is 25.1 cm, double, but the area is 150.8 cm², four times as much. Length scales with the radius and area with its square.
  3. Keep radius 6 and double the angle to 240°: the arc is 25.1 cm and the area is 75.4 cm², both double, because the fraction of the circle doubles.
  4. Set radius 4 and angle 300°: the fraction is 5/6, the arc is 20.9 cm and the area is 41.9 cm². The angle can be more than 180°; the same fraction rule holds.

A sector is the fraction t/360 of a circle. Its arc is that fraction of the circumference: arc = (t/360) × 2πr, and its area is that fraction of the area: (t/360) × πr². Its perimeter is the arc plus two radii.

Length scales with r, area with r². A segment is a sector minus a triangle: area = (t/360)πr² − ½r² sin t.

8
90
50.3 cm²area of the sector
18.3 cm²area of the segment

Radius 8, angle 90°: sector 50.3, triangle ½ × 8² × sin 90° = 32.0, segment 50.3 − 32.0 = 18.3 cm².

Show me the steps
  1. Radius 8, angle 90°: the sector is a quarter circle, 50.3. The triangle is ½ × 64 × sin 90° = 32.0. The segment is 50.3 − 32.0 = 18.3 cm².
  2. Keep radius 8 and set the angle to 60°: the sector is 33.5, the triangle ½ × 64 × sin 60° = 27.7, and the segment is 5.8. A small angle leaves a thin segment.
  3. Set the angle to 180°: the triangle has no area (sin 180° = 0) because the chord is a diameter, so the segment is the whole sector, a half circle of 100.5 cm².
  4. Set the radius to 10 and the angle to 150°: the sector is 130.9, the triangle 25.0, and the segment 105.9. The triangle can take only a little of a wide sector.

The perimeter of a sector includes the two straight edges. A sector with radius 5 cm and an arc of 8 cm has a perimeter of 8 + 5 + 5 = 18 cm, not 8 cm. Say which part of the outline each question asks for before you start.

Worked examples

1. A sector has radius 12 cm and angle 105°. Find its arc length and area, in terms of π and to 1 decimal place.

  1. The fraction is 105 ÷ 360 = 7/24.
  2. Arc = 7/24 × 2π × 12 = 7π = 22.0 cm.
  3. Area = 7/24 × π × 144 = 42π = 131.9 cm².

2. Find the area of the segment cut off by a chord that makes 60° at the centre of a circle of radius 10 cm, exactly and to 1 decimal place.

  1. Sector: 60/360 × π × 100 = 50π/3.
  2. Triangle: ½ × 100 × sin 60° = 25√3, because the triangle is equilateral.
  3. Segment = 50π/3 − 25√3 = 9.1 cm².

3. Show that the area of a sector is ½ × arc × r.

  1. Arc = (t/360) × 2πr and area = (t/360) × πr².
  2. ½ × arc × r = ½ × (t/360) × 2πr × r = (t/360) × πr², which is the area.
  3. So a sector behaves like a triangle whose base is the arc and whose height is the radius.

Your go. The first step is given. A sector has radius 18 cm and angle 50°. Find its area in terms of π, and then to 1 decimal place.

  1. The fraction is 50 ÷ 360 = 5/36.
  2. Multiply by π × 18² and cancel before you use a calculator.
Check the answer

5/36 × 324π = 45π ≈ 141.4 cm². Using the diameter, 36, in place of the radius would multiply the area by 4.

With a GDC (optional)

Use the π key rather than 3.14, and enter a fraction of a circle with brackets, such as (105 ÷ 360) × 2 × π × 12. Press equals once at the end. For an answer in terms of π, work out the number in front by hand (here 7) and write 7π; the calculator is then only a check on the decimal.

To find a segment, store the sector, work out half r² sin θ with the calculator in degrees, and subtract. Everything here can be done with a scientific calculator.

Check before you go on

1. Sector P has radius 6 cm and angle 80°. Sector Q has radius 12 cm and angle 40°. Which statement is true?

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 of a circle has radius 15 cm and an angle of 54° at the centre. What is the length of its arc, to 1 decimal place?

3. A sector of a circle has radius 10 cm and an angle of 252° at the centre. What is its area, to 1 decimal place?

Level 2 · Building

4. A sector has an angle of 75° at the centre and its arc is 20 cm long. What is the radius, to 1 decimal place?

5. A sector has radius 7 cm and an angle of 150° at the centre. What is the perimeter of the sector, including its two straight edges, to 1 decimal place?

6. A sector of a circle of radius 8 cm has an area of 40 cm². What is the angle at the centre, to the nearest degree?

Level 3 · Stretch

7. A chord of a circle of radius 9 cm makes an angle of 100° at the centre. Find the area of the smaller segment cut off by the chord, to 1 decimal place.

8. A flower bed is the part of a circular lawn between a circle of radius 8 m and a circle of radius 14 m with the same centre, cut off by two radii at 75° to each other. Find its area, to 1 decimal place.

Something to think about

Keep the area of a sector fixed and double its radius. What must happen to the angle, and what happens to the arc? Try it with radius 6 cm and angle 80°, then radius 12 cm.

One way to think about it

The angle must be divided by 4, because the area depends on r² × angle: 80° becomes 20°. The arc depends on r × angle, so it is multiplied by 2 and divided by 4: it halves. Radius 6 and 80° give area 8π and arc 8π/3; radius 12 and 20° give area (20/360) × 144π = 8π and arc (20/360) × 24π = 4π/3.

Exit ticket

9. A quarter of a circle (a sector with a 90° angle) has an arc of length 5π cm. What is the radius of the circle?