Mathematics • Year 10 • Unit 1 • Lesson 15
Composite Areas in the Real World
Apply Lesson 15's addition/subtraction strategies to real Australian measurement scenarios — surveying suburban blocks, costing carpet, ordering tiles, designing a deck with a spa. Then explain in your own words how to choose between the two strategies.
1. Word problems
Each problem requires you to choose the right strategy (addition vs subtraction), pick the right area formulas, and apply unit consistency.
1.1 — Costing carpet for an L-shaped room. A floor plan consists of a rectangle 12 m by 8 m joined to a trapezium with parallel sides 8 m and 5 m, height 3 m. There is also a square structural column 1 m by 1 m to be excluded from the carpet.
(a) Calculate the rectangle area.
(b) Calculate the trapezium area.
(c) Calculate the total carpet area, accounting for the column.
(d) Carpet costs $45 per m². What is the total cost? 4 marks
1.2 — Deck around a spa. A rectangular outdoor deck measures 15 m by 10 m. A circular spa of radius 2.5 m is built into the deck.
(a) Find the area of the deck surrounding the spa, to 1 decimal place. (Use π ≈ 3.1416.)
(b) Decking timber is sold by the square metre. If 5 % wastage is added, how many m² of timber should be ordered (rounded up to the nearest m²)?
(c) Why is the subtraction strategy more efficient than the addition strategy for this shape? 3 marks
1.3 — Garden bed shaped like a house. A garden bed is a rectangle 8 m by 5 m with a right-angled triangle of base 5 m and height 3 m on one of the 5 m sides (like a "house" outline).
(a) Find the total area of the garden bed.
(b) Garden mulch is sold at $12 per m². Calculate the total cost to mulch the garden bed. 3 marks
1.4 — Tiling around a circular skylight. A ceiling measures 6 m by 4 m. A circular skylight of diameter 1.6 m is cut into it. The remaining ceiling area is to be tiled.
(a) Calculate the radius of the skylight.
(b) Find the area of the skylight in terms of π.
(c) Calculate the area to be tiled (use π ≈ 3.1416, give answer to 2 decimal places). 3 marks
1.5 — Suburban land block. Lesson 15's Real-Life Link mentions surveyors measuring blocks where the front boundary is curved (a sector). A suburban block is a 25 m × 20 m rectangle with a sector of radius 5 m (angle 90°) cut from one corner for a curved driveway.
(a) Calculate the rectangle area.
(b) Calculate the sector area in terms of π, then to 1 decimal place.
(c) Calculate the total block area (rectangle minus sector). 3 marks
2. Explain your thinking
This question is about communication, not just numbers. Use full sentences. 4 marks
2.1 A friend says: "I always use the addition strategy — just split everything into rectangles and triangles, then add. Why would anyone ever subtract?" Using everything from Lesson 15, explain (i) when the subtraction strategy is genuinely faster and more accurate (give one example from Section 1), (ii) what goes wrong if you try to split a rectangle-with-a-circular-hole using only addition, and (iii) why the lesson says "choosing the most efficient dissection saves time and reduces error". Refer to "enclosing shape" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — L-shaped room with column
(a) Rectangle = 12 × 8 = 96 m².
(b) Trapezium = ½(8 + 5) × 3 = ½ × 13 × 3 = 19.5 m².
(c) Combined = 96 + 19.5 = 115.5 m². Subtract column: 115.5 − 1 = 114.5 m².
(d) Cost = 114.5 × $45 = $5,152.50.
1.2 — Deck around spa
(a) Deck area = 15 × 10 = 150 m². Spa area = π × 2.5² = 6.25π m² ≈ 19.6 m². Surrounding = 150 − 19.6 ≈ 130.4 m².
(b) With 5 % wastage: 130.4 × 1.05 ≈ 136.92 m² → order 137 m².
(c) The deck minus the spa is an irregular shape with curved boundaries; splitting it into rectangles and circular sectors using the addition strategy would need many parts, each a potential source of error. Subtracting one circle from one rectangle takes just two steps.
1.3 — House-shaped garden bed
(a) Rectangle = 8 × 5 = 40 m². Triangle = ½ × 5 × 3 = 7.5 m². Total = 47.5 m².
(b) Cost = 47.5 × $12 = $570.00.
1.4 — Ceiling with skylight
(a) r = d/2 = 1.6 / 2 = 0.8 m.
(b) Skylight area = π × 0.8² = 0.64π m².
(c) Ceiling = 6 × 4 = 24 m². Tile area = 24 − 0.64π ≈ 24 − 2.0106 ≈ 21.99 m² (2 d.p.).
1.5 — Land block with curved driveway
(a) Rectangle = 25 × 20 = 500 m².
(b) Sector area = (90/360) × π × 5² = (1/4) × 25π = 6.25π m² ≈ 19.6 m².
(c) Block area = 500 − 6.25π ≈ 500 − 19.6 ≈ 480.4 m².
This is exactly the kind of suburban-block calculation the lesson's Real-Life Link describes.
2.1 — Explain your thinking (sample response)
(i) Subtraction is faster when the composite is a basic shape with a hole or a missing corner — for example Q1.2's deck minus a circular spa, where the leftover region is too irregular to add up cleanly. (ii) If you tried to use addition on a rectangle with a circular hole, you'd have to dissect the irregular outer region into many sectors and partial rectangles, each requiring careful angle and dimension calculations, and each adding rounding error to the total. (iii) The lesson states "choosing the most efficient dissection saves time and reduces error" because fewer parts means fewer formulas, fewer multiplications, and fewer chances for sign errors or missed pieces — finding the area of the smallest enclosing shape and subtracting the missing part is often a two-step calculation that replaces a fifteen-step addition.
Marking: 1 for naming when subtraction is faster, 1 for an example from Section 1, 1 for explaining the error-amplification problem of over-dissection, 1 for using "enclosing shape" correctly.