Three new area formulas — all built on the circle. Master the sector, the ring, and the triangle with an included angle.
55–60 minMS-M1 — HIGH3 MC3 SALesson 6 of 22Free
🔵
Choose how you work: type answers on screen, or work in your book.
Think First
A circular garden has a circular pond in the centre. You want to turf the garden but not the pond. You know both radii. How would you find the area to turf — and why can't you just measure it directly?
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.
Write your initial thinking in your book
Saved
Come back to this at the end of the lesson.
📋
Key Formulas — This Lesson
$A = \dfrac{\theta}{360} \times \pi r^2$
Sector area — $\theta$ = central angle, $r$ = radiusThink: fraction $\theta/360$ of the full circle area $\pi r^2$
$A = \pi(R^2 - r^2)$
Annulus area — $R$ = outer radius, $r$ = inner radius ($R > r$)Outer circle area minus inner circle area
$A = \tfrac{1}{2}ab\sin C$
Sine area rule — $a$, $b$ = two known sides, $C$ = included angle between themUsed when perpendicular height is unknown; $C$ must be between sides $a$ and $b$
Arc length vs sector area: Both use $\theta/360$ — arc length multiplies by $2\pi r$, sector area multiplies by $\pi r^2$
Know
The sector area formula $A = (\theta/360) \times \pi r^2$
The annulus area formula $A = \pi(R^2 - r^2)$
The sine area rule $A = \tfrac{1}{2}ab\sin C$
Understand
Why sector area is a fraction of $\pi r^2$ — same fraction as arc length uses on circumference
Why annulus area = outer area − inner area
Why $A = \tfrac{1}{2}ab\sin C$ works when no perpendicular height is given
Can Do
Calculate sector, annulus, and triangle areas using the correct formula
Identify when each formula applies
Solve composite area problems combining these formulas with L02 shapes
🔑
Key Vocabulary
SectorA region bounded by two radii and an arc — a "pizza slice" of a circle
AnnulusThe region between two concentric circles — a "ring" shape
Concentric circlesTwo or more circles with the same centre but different radii
Included angleThe angle formed between two known sides of a triangle — the angle sitting between them
Exact formLeaving an answer as a multiple of $\pi$ (e.g. $18\pi$ cm²) — used when a question says "leave in terms of $\pi$"
Misconceptions to Fix
✗
Wrong: Sector area = (theta/360) x pi x r^2 without the 1/2 factor.
✓
Right: The correct formula is A = (theta/360) * pi * r² for degrees, or A = 0.5 * r² * theta for radians.
Core Content
01
Area of a Sector
A sector is a fraction of a full circle. The fraction is $\theta/360$. So the sector area is that same fraction of $\pi r^2$.
$$A = \frac{\theta}{360} \times \pi r^2$$
Formula
What follows $\theta/360$
Arc length $\ell$
Circumference: $2\pi r$
Sector area $A$
Full circle area: $\pi r^2$
Comparing the two formulas: The fraction $\theta/360$ is identical in both. The only difference is whether you multiply by $2\pi r$ (circumference) or $\pi r^2$ (area). If you can keep this comparison in mind, you will never mix up the two formulas.
Unit check: Arc length → linear units (cm, m). Sector area → square units (cm², m²). If your "arc length" answer has a² units, you used the wrong formula. Always check units.
✏️
Sector, Annulus, and Composite Area Practice
Label which formula you are using before substituting.
02
Area of an Annulus
An annulus is the ring between two concentric circles. Its area = outer circle area − inner circle area.
$$A = \pi R^2 - \pi r^2 = \pi(R^2 - r^2)$$
Both forms are correct. The factored form $\pi(R^2 - r^2)$ is more efficient on a calculator — compute the bracket first, then multiply by $\pi$ once.
Label clearly: $R$ = outer (larger) radius, $r$ = inner (smaller) radius. Always identify and write both before substituting. The subtraction is always $R^2 - r^2$ — subtracting in the wrong order gives a negative area.
Rounding tip: Compute $R^2 - r^2$ first (exact), then multiply by $\pi$ at the final step. Avoid computing $\pi R^2$ and $\pi r^2$ separately — this introduces two rounding errors instead of one.
✏️
03
The Sine Area Rule
The basic formula $A = \tfrac{1}{2}bh$ needs the perpendicular height. When two sides and the included angle are given instead, use the sine area rule.
$$A = \frac{1}{2}ab\sin C$$
where $a$ and $b$ are two known side lengths and $C$ is the included angle — the angle sitting between those two sides.
When $C = 90°$
$\sin 90° = 1$, so the formula becomes $A = \tfrac{1}{2}ab$ — consistent with a right-angled triangle where the two sides are the base and height. The sine rule reduces to the standard formula as a special case.
The included angle check: Before applying $A = \tfrac{1}{2}ab\sin C$, confirm that $C$ is at the vertex where sides $a$ and $b$ meet. If the angle is at a different vertex, it is not the included angle for those sides and the formula gives the wrong answer.
Check Your Understanding
Write one sentence summarising the main mathematical idea of this section.
04
Composite Areas — Combining All Formulas
The strategy from L02 still applies: identify components, decide add or subtract, calculate each, combine.
Common combination
Method
Rectangle with sector cut from corner
Rectangle area − sector area
Triangle with sector attached
Triangle area + sector area
Circle with hole (= annulus)
$\pi(R^2 - r^2)$ directly
Logo with three corner sectors removed
Triangle area − 3 × sector area
Negative area = wrong decision: If you get a negative area, you have added when you should have subtracted (or the component is actually larger than the shape it is being removed from). Never write a negative area as a final answer — use it as a signal to re-examine your add/subtract choice.
Check Your Understanding
Write down the key formula from this section and explain what each variable represents.
05
Common Mistakes
Mistake 1 — Arc length formula instead of sector area
Using $(θ/360) \times 2\pi r$ when asked for area gives a length answer. Check units: area must have square units. If you get cm when the question asks for cm², you used the arc formula.
Mistake 2 — Wrong subtraction order in annulus
$\pi(r^2 - R^2)$ gives a negative area. Always write $R$ = outer and $r$ = inner before substituting, then subtract $R^2 - r^2$ (larger minus smaller).
Mistake 3 — Non-included angle in sine area rule
Using an angle that is not between the two known sides gives a wrong area even though the formula looks correct. The included angle $C$ must be at the vertex where sides $a$ and $b$ meet.
Check Your Understanding
Write one sentence summarising the main mathematical idea of this section.
Worked Examples
06
Worked Example 1
Area of a Sector
Problem
Find the area of a sector with radius 10 cm and central angle 144°. Give your answer correct to 2 decimal places.
Step-by-Step Solution
1
Write formula and substitute $A = \dfrac{144}{360} \times \pi \times 10^2$
Sector area formula. $\theta = 144°$, $r = 10$ cm.
1 Find the area of a sector with $r = 5$ cm and $\theta = 72°$. Give in exact form and to 2 decimal places.
Working space in book
Saved
2 Find the area of a sector with $r = 12$ m and $\theta = 150°$. Answer to 2 decimal places.
Working space in book
Saved
3 A sector has area $30\pi$ cm² and radius 6 cm. Find the central angle $\theta$.
Working space in book
Saved
Section B — Annulus Area
4 Find the area of an annulus with $R = 10$ cm and $r = 6$ cm. Answer to 2 decimal places.
Working space in book
Saved
5 A circular path surrounds a garden of radius 4 m. The path extends 1.5 m beyond the garden edge. Find the area of the path to 2 decimal places.
Working space in book
Saved
Section C — Sine Area Rule
6 Find the area of a triangle with sides 7 cm and 9 cm and included angle 40°. Answer to 2 decimal places.
Working space in book
Saved
7 Find the area of a triangle with sides 15 m and 15 m and included angle 100°. Answer to 2 decimal places.
Working space in book
Saved
Section D — Composite Areas
8 A square of side 8 cm has a sector of radius 8 cm and angle 90° removed from one corner. Find the remaining area to 2 decimal places.
Working space in book
Saved
9 A shape consists of a rectangle 10 cm × 6 cm, with a semicircle of diameter 6 cm added to one short end and a triangle (base 6 cm, two sides 6 cm each, included angle 50°) removed from the other short end. Find the total area to 2 decimal places.
2 An annulus has outer radius 8 m and inner radius 3 m. Its area, correct to 2 decimal places, is:
A $25\pi$ m²
B $55\pi$ m²
C 78.54 m²
D 172.79 m²
D — $A = \pi(64 - 9) = 55\pi = 172.79$ m² (to 2 d.p.). Option B is the exact form — both B and D are correct representations, but D gives the decimal answer.
3 A triangle has two sides of 10 cm and 14 cm with an included angle of 30°. Its area is:
SA 42 marks
Find the area of a sector with radius 9 m and central angle 200°. Give your answer correct to 2 decimal places.
Working space in book
Saved
11
SA 53 marks
A circular table has diameter 1.6 m. A circular lazy Susan with diameter 0.8 m sits in the centre. Find the area of the table not covered by the lazy Susan, correct to 2 decimal places.
Working space in book
Saved
12
SA 64 marks
A logo is made from an equilateral triangle with side length 8 cm. A sector of radius 3 cm and central angle 60° is drawn at each vertex.
(a) Find the area of the equilateral triangle using the sine area rule, correct to 2 decimal places. (2 marks)
(b) Find the total area of the three sectors. (1 mark)
(c) Find the area of the logo (triangle minus sectors), correct to 2 decimal places. (1 mark)