Mathematics Standard • Year 11 • Module 2 • Lesson 2
Area of Basic Shapes
Build fluency with the five core area formulas — rectangle, triangle, parallelogram, trapezium, circle — then add composite shapes.
1. Quick recall — area formulas
Fill in each formula. 1 mark each
Q1.1 Complete each area formula (use ℓ, w, b, h, a, r as needed):
Rectangle: A = __________ Triangle: A = __________ Parallelogram: A = __________
Trapezium: A = __________ Circle: A = __________
Q1.2 In the formulas above, what does h mean? Circle the correct answer.
(a) the slant side (b) any side (c) the perpendicular height (at 90° to the base)
Q1.3 A circle has diameter d. Write the radius r in terms of d:
r = ____________
2. Worked example — area of a circle from diameter
Follow each line of working — every step earns a method mark.
Problem. Find the area of a circle with diameter 14 cm. Answer correct to 2 decimal places.
Step 1 — Halve the diameter to find the radius.
r = 14 ÷ 2 = 7 cm
Reason: write this as a separate line — the most common error is substituting d directly into r².
Step 2 — Write the formula and substitute.
A = πr² = π × 7² = π × 49
Reason: keep π exact at this stage — do not evaluate yet.
Step 3 — Evaluate using the π button.
A = 153.9380...
Reason: use the calculator π button — never 3.14. Round only at the very end.
Step 4 — Round and state the answer with units.
A = 153.94 cm²
Reason: the question asks for 2 d.p. If you had used d = 14 in r²: A = π × 196 = 615.75 cm² — four times too large.
3. Faded example — area of a trapezium
A trapezium has parallel sides 5 m and 11 m, and perpendicular height 4 m. Find its area. Fill in every blank. 3 marks
Step 1 — Identify and label: a = ______, b = ______, h = ______.
Step 2 — Write the formula: A = ½ (______ + ______) × ______.
Step 3 — Brackets first: A = ½ × ______ × 4.
Step 4 — Evaluate: A = ______ m².
Common slip: forgetting the ½ gives 64 — double the correct answer.
4. Graduated practice — calculate the area
For every question: write the formula, substitute, then state the answer with the correct unit.
Foundation — single shape, clean numbers (4 questions)
| Q | Problem | Answer (with unit) |
|---|---|---|
| 4.1 1 | Rectangle: ℓ = 13 cm, w = 7 cm. Find A. | |
| 4.2 1 | Triangle: b = 10 m, h = 6 m. Find A. | |
| 4.3 1 | Parallelogram: b = 8 cm, h = 5 cm. Find A. | |
| 4.4 1 | Trapezium: a = 4 cm, b = 9 cm, h = 6 cm. Find A. |
Standard — typical HSC difficulty (6 questions)
4.5 Circle: r = 5 cm. Find A to 2 d.p. 2 marks
4.6 Circle: diameter 18 m. Find A to 2 d.p. (Show r as a separate step.) 2 marks
4.7 Triangle: b = 12 m, slant side = 8 m, perpendicular height = 6 m. Find A. (Beware: the slant side is a distractor.) 2 marks
4.8 Trapezium: a = 7 cm, b = 13 cm, h = 8 cm. Find A. 2 marks
4.9 Composite — addition: a rectangle (8 cm × 5 cm) with a triangle (b = 8 cm, h = 3 cm) sitting on top. Find the total area. 3 marks
4.10 Composite — subtraction: a square (side 10 m) with a circle (r = 3 m) removed from the centre. Find the remaining area to 2 d.p. 3 marks
Extension — multi-step composite shapes (2 questions)
4.11 A rectangle of timber is 20 cm long and 12 cm wide. A semicircle with diameter equal to the rectangle's width is cut from one end. Find the area of the remaining shape to 2 d.p. 3 marks
4.12 A running track is a rectangle 80 m × 40 m with a semicircle on each short end. Find the total enclosed area to 1 d.p. (Hint: two semicircles of equal radius = one full circle.) 3 marks
5. Self-check the easy 3
Tick the first three once you've checked your method.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — The five area formulas
Rectangle: A = ℓw. Triangle: A = ½bh. Parallelogram: A = bh. Trapezium: A = ½(a + b)h. Circle: A = πr².
Q1.2 — What is h?
(c) the perpendicular height (measured at 90° to the base). Never the slant side.
Q1.3 — Radius from diameter
r = d ÷ 2.
Q3 — Faded trapezium
Step 1: a = 5, b = 11, h = 4. Step 2: A = ½(5 + 11) × 4. Step 3: A = ½ × 16 × 4. Step 4: A = 32 m².
Q4.1 — Rectangle
A = 13 × 7 = 91 cm².
Q4.2 — Triangle
A = ½ × 10 × 6 = 30 m².
Q4.3 — Parallelogram
A = 8 × 5 = 40 cm².
Q4.4 — Trapezium
A = ½(4 + 9) × 6 = ½ × 13 × 6 = 39 cm².
Q4.5 — Circle, r = 5
A = π × 5² = 25π = 78.54 cm² (to 2 d.p.).
Q4.6 — Circle, d = 18 m
r = 18 ÷ 2 = 9 m. A = π × 81 = 254.47 m² (to 2 d.p.).
Q4.7 — Triangle with distractor slant
A = ½ × 12 × 6 = 36 m². The 8 m slant side plays no role in the area formula.
Q4.8 — Trapezium
A = ½(7 + 13) × 8 = ½ × 20 × 8 = 80 cm².
Q4.9 — Rectangle + triangle (addition)
Plan: A = rectangle + triangle. Rectangle = 8 × 5 = 40 cm². Triangle = ½ × 8 × 3 = 12 cm². Total = 52 cm².
Q4.10 — Square − circle (subtraction)
Plan: A = square − circle. Square = 10² = 100 m². Circle = π × 3² = 9π = 28.274... m². Remaining = 100 − 28.27 = 71.73 m² (to 2 d.p.).
Q4.11 — Rectangle − semicircle
Plan: A = rectangle − semicircle. Rectangle = 20 × 12 = 240 cm². Semicircle: r = 12 ÷ 2 = 6, so Asemi = ½ × π × 36 = 18π = 56.548... cm². Remaining = 240 − 56.55 = 183.45 cm² (to 2 d.p.).
Q4.12 — Running track (rectangle + two semicircles)
Two semicircles of equal radius combine into one full circle. r = 40 ÷ 2 = 20 m. Rectangle = 80 × 40 = 3200 m². Circle = π × 20² = 400π = 1256.637... m². Total = 3200 + 1256.6 = 4456.6 m² (to 1 d.p.).