Circumference \(C = 2\pi r\) and area \(A = \pi r^2\) are given in the List of formulas. For a sector with angle \(\theta\): arc length \(= \dfrac{\theta}{360} \times 2\pi r\) and sector area \(= \dfrac{\theta}{360} \times \pi r^2\). These two are NOT printed: build them from the fraction of the circle.
Fig. 1: a 60 degree sector of a circle of radius 6 cm is one sixth of the circle, so its arc is one sixth of the circumference and its area one sixth of the circle's area
Section 01
Circumference and area.
In the lab, change the radius and press Roll it to see the circumference unroll into a straight line just over 3 diameters long. Then press Rearrange: the slices of the circle make a near-rectangle of width \(\pi r\) and height \(r\), which is why the area is \(\pi r^2\).
Given in the List of formulas: \(C = 2\pi r\) and \(A = \pi r^2\).
Radius or diameter? \(r\) is the radius. If you are given the diameter \(d\), halve it first (or use \(C = \pi d\)).
"In terms of \(\pi\)" means leave \(\pi\) as a symbol: \(9\pi\), not 28.3.
Otherwise use the calculator's \(\pi\) key and give 3 significant figures.
Worked example From the radius
A circle has radius 4.5 cm. Find (a) its circumference (b) its area.
A circle has circumference 50 cm. Find its radius.
\(2\pi r = 50\), so \(r = 50 \div (2\pi) = 7.957... = 7.96\) cm.
Examiner trap
Putting the diameter into \(\pi r^2\): a circle of diameter 10 cm has area \(25\pi\), not \(100\pi\). Halve the diameter before you square.
Section 02
Arcs and sectors.
In the lab, drag the red point to change the sector angle and watch the fraction \(\theta/360\) change. A 60° sector is \(\tfrac{60}{360} = \tfrac{1}{6}\) of the circle (Fig. 1), so it gets one sixth of the circumference and one sixth of the area.
Worked example Arc length and sector area
A sector has radius 6 cm and angle 60°. Find (a) its arc length (b) its area. Give your answers in terms of \(\pi\).
Fraction of the circle: \(\dfrac{60}{360} = \dfrac{1}{6}\).
Answer: Perimeter \((5\pi + 20)\) cm, area \(25\pi\text{ cm}^2\)
Your turn
Find the perimeter of a semicircle of diameter 12 cm.
Arc \(= \tfrac{1}{2} \times 2\pi \times 6 = 6\pi\). Perimeter \(= 6\pi + 12 = 30.8\) cm (the straight edge is the diameter).
Examiner trap
The perimeter of a sector is NOT just the arc. Add the two radii (for a semicircle, the diameter). Leaving them out is the examiner's favourite lost mark.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Which of these is printed in the 0580 List of formulas?
Quick check+10 XP
Find the circumference of a circle of diameter 8 cm, in terms of \(\pi\).
Quick check+10 XP
What fraction of a circle is a sector with angle 45°?
Quick check+10 XP
Find the area of a circle of radius 3 cm, in terms of \(\pi\).
Section 03
Any angle, and major sectors.
Extended candidates meet any sector angle, the larger (major) sector, and questions that give the arc or area and ask for the angle or radius. The fraction method still works: write the equation and solve it.
Worked example Major sector (Extended)
A circle of radius 8 cm has a minor sector of angle 135°. Find (a) the minor arc length (b) the area of the major sector.