Trace the boundary. Every edge counts — straight or curved. The arc is just a fraction of the full circumference, and the fraction is determined by the angle.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
A pizza slice (sector) has a curved crust and two straight edges going in to the centre. If you wanted to know the total length of pastry needed to frame the slice, which edges would you measure? Is the crust the only part that matters?
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Come back to this at the end of the lesson.
Wrong: Rounding 4.5 down gives 4.
Right: Standard rounding rounds 4.5 up to 5; rounding to the nearest even number (banker's rounding) is different.
Core Content
Perimeter is the total length of the boundary. For any polygon — add every side. For composite shapes — trace the outer boundary and add every edge you cross.
| Shape | Formula |
|---|---|
| Rectangle | $P = 2\ell + 2w$ |
| Square | $P = 4s$ |
| Triangle | $P = a + b + c$ |
| Circle | $C = 2\pi r = \pi d$ |
An arc is a curved portion of a circle's circumference. A sector with central angle $\theta$ contains an arc that is exactly $\theta/360$ of the full circumference.
$$\ell = \frac{\theta}{360} \times 2\pi r$$| $\theta$ | Fraction of circle | Arc length |
|---|---|---|
| 360° | Full circle | $2\pi r$ — the full circumference ✓ |
| 180° | Semicircle | $\pi r$ |
| 90° | Quarter circle | $\tfrac{1}{2}\pi r$ |
| 60° | Sixth of circle | $\tfrac{1}{3}\pi r$ |
A sector has three edges — two straight radii and one curved arc. The perimeter includes all three.
$$P = 2r + \ell = 2r + \frac{\theta}{360} \times 2\pi r$$Many HSC problems combine straight sides and arcs. Trace the boundary, identify each edge, calculate each separately, then add.
A rectangle with a semicircle on each short end. The outer perimeter consists of:
Worked Examples
Find the arc length of a sector with radius 9 cm and central angle 80°. Give your answer correct to 2 decimal places.
Find the perimeter of a sector with radius 12 m and central angle 135°. Give your answer correct to 2 decimal places.
A running track consists of a rectangle 60 m long and 20 m wide, with a semicircle attached to each short end. Find the perimeter of the outside edge of the track correct to 2 decimal places.
A shape is formed by taking a square of side 10 cm and attaching a sector of radius 10 cm and angle 60° to one side. Find the perimeter of the composite shape correct to 2 decimal places.
For composite shapes, describe which edges form the boundary before calculating.
Section A — Arc Length
1 Find the arc length of a sector with $r = 6$ cm and $\theta = 90°$.
2 Find the arc length of a sector with $r = 15$ m and $\theta = 120°$.
3 Find the arc length of a sector with $r = 8$ cm and $\theta = 45°$. Answer to 2 decimal places.
Section B — Perimeter of a Sector
4 Find the perimeter of a sector with $r = 10$ cm and $\theta = 90°$.
5 Find the perimeter of a sector with $r = 7$ m and $\theta = 150°$. Answer to 2 decimal places.
Section C — Composite Perimeter
6 A shape is made from a rectangle 8 cm × 5 cm with a semicircle of diameter 5 cm attached to one short end. Find the perimeter of the outside edge to 2 decimal places.
7 A sector has radius 6 m and angle 240°. Find its perimeter to 2 decimal places.
8 A quarter-circle of radius 4 cm is removed from the corner of a square of side 4 cm. Find the perimeter of the resulting shape to 2 decimal places.
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
Multiple Choice
1 A sector has radius 5 cm and central angle 72°. Its arc length is:
? Regarding this topic, 1 A sector has radius 5 cm and central angle 72°. Its arc length is:
2 A sector has radius 8 m and arc length 12 m. Its perimeter is:
? Regarding this topic, 2 A sector has radius 8 m and arc length 12 m. Its perimeter is:
3 A rectangle (14 cm × 6 cm) has a semicircle of diameter 6 cm removed from one short end. The perimeter of the remaining shape is closest to:
? Regarding this topic, 3 A rectangle (14 cm × 6 cm) has a semicircle of diameter 6 cm removed from one short end. The perimeter of the remaining shape is closest to:
Short Answer
SA 4 2 marks Find the arc length of a sector with radius 18 cm and central angle 40°. Give your answer correct to 2 decimal places.
SA 5 3 marks Find the perimeter of a sector with radius 14 cm and central angle 225°. Give your answer correct to 2 decimal places.
SA 6 4 marks A garden path is bounded by three straight sides (8 m, 5 m, and 8 m) and a curved arc on the fourth side. The arc is part of a circle with radius 5 m, centred at the midpoint of the 5 m side.
(a) Find the central angle of the arc. (1 mark)
(b) Find the arc length correct to 2 decimal places. (1 mark)
(c) Find the total perimeter correct to 2 decimal places. (2 marks)
Face the boss using your knowledge of perimeter and arc length calculations. Pool: lessons 1–5.
Tick when you have finished the lesson and checked your answers.