Mathematics • Year 10 • Unit 1 • Lesson 15
Area of Composite Shapes — Skill Drill
Build fluency with Lesson 15's two strategies — addition (dissect into basic shapes and add) and subtraction (find an enclosing shape, subtract the missing part). Practise the six core area formulas first, then combine them in graduated problems.
1. I do — fully worked example
Read every step. Each one has a short reason on the right so you can see why, not just what.
Problem. An L-shaped room consists of a rectangle 8 m by 5 m joined to a rectangle 4 m by 3 m. Find the total floor area.
Step 1 — Choose the strategy.
The shape is made of two rectangles joined together → use the ADDITION strategy.
Reason: from the Addition Strategy card — when a shape splits cleanly into basic shapes, dissect and add.
Step 2 — Dissect into basic shapes.
Part 1: rectangle 8 m × 5 m.
Part 2: rectangle 4 m × 3 m.
Reason: the room is already labelled as two rectangles, so no missing-length calculation needed.
Step 3 — Apply the rectangle formula A = l × w to each part.
Area 1 = 8 × 5 = 40 m²
Area 2 = 4 × 3 = 12 m²
Step 4 — Add the areas.
Total area = 40 + 12 = 52 m²
Answer: Total area = 52 m².
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. A rectangular garden 12 m by 8 m has a circular pond of radius 2 m in the centre. Find the area of grass surrounding the pond.
Step 1 — Choose the strategy. A shape with a hole inside is best handled by the __________ strategy.
Step 2 — Find the enclosing area (the whole rectangle):
Rectangle area = ______ × ______ = __________ m²
Step 3 — Find the missing part (the pond), using A = π r²:
Pond area = π × ______² = __________ π m²
Step 4 — Subtract.
Grass area = __________ − __________ π m²
Step 5 — Evaluate to 1 decimal place (using π ≈ 3.1416):
Grass area ≈ __________ m²
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (single basic-shape formula). The middle two are standard (addition / subtraction). The last two are extension (multi-part composite).
Foundation — single basic-shape formulas
3.1 Calculate the area of a triangle with base 12 cm and perpendicular height 5 cm. 1 mark
3.2 Calculate the area of a parallelogram with base 9 m and perpendicular height 4 m. 1 mark
3.3 Calculate the area of a trapezium with parallel sides 8 m and 5 m and perpendicular height 3 m. 1 mark
3.4 Calculate the area of a sector with radius 6 cm and angle 60°. Leave your answer in terms of π. 1 mark
Standard — addition / subtraction strategies
3.5 A composite shape is made from a rectangle 10 cm by 6 cm with a triangle of base 6 cm and height 4 cm on top. Find the total area using the addition strategy. 2 marks
3.6 A square of side 10 cm has a smaller square of side 4 cm cut out from one corner. Find the remaining area using the subtraction strategy. 2 marks
Extension — push your thinking
3.7 A composite shape is a rectangle 9 m by 4 m with a semicircle of diameter 4 m on one end. Find the total area in terms of π. 3 marks
3.8 A square patio of side 7 m has a rectangular planter of 2 m by 1.5 m removed from one corner. Find the remaining patio area and state which strategy you used (addition or subtraction). 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded 12 × 8 garden with pond)
Step 1: subtraction strategy.
Step 2: Rectangle area = 12 × 8 = 96 m².
Step 3: Pond area = π × 2² = 4π m².
Step 4: Grass area = 96 − 4π m².
Step 5: Grass area ≈ 96 − 12.57 ≈ 83.4 m² (1 d.p.).
3.1 — Triangle
A = ½ × b × h = ½ × 12 × 5 = 30 cm².
3.2 — Parallelogram
A = b × h = 9 × 4 = 36 m². (Note: must use the perpendicular height, not the side length — Misconceptions card.)
3.3 — Trapezium
A = ½(a + b) × h = ½(8 + 5) × 3 = ½ × 13 × 3 = 19.5 m².
3.4 — Sector (60°, radius 6 cm)
A = (θ/360) × π r² = (60/360) × π × 36 = (1/6) × 36π = 6π cm².
3.5 — Rectangle + triangle on top
Rectangle = 10 × 6 = 60 cm². Triangle = ½ × 6 × 4 = 12 cm². Total = 60 + 12 = 72 cm².
3.6 — Square with square cut-out
Big square = 10 × 10 = 100 cm². Small square = 4 × 4 = 16 cm². Remaining = 100 − 16 = 84 cm².
3.7 — Rectangle + semicircle
Rectangle = 9 × 4 = 36 m². Semicircle radius = 4 ÷ 2 = 2 m. Semicircle area = ½ × π × 2² = ½ × 4π = 2π m².
Total = (36 + 2π) m² (≈ 42.3 m²).
3.8 — Square patio with planter cut out
Strategy: subtraction.
Patio = 7 × 7 = 49 m². Planter = 2 × 1.5 = 3 m². Remaining = 49 − 3 = 46 m².