Mathematics Standard • Year 11 • Module 2 • Lesson 2

Area of Basic Shapes

Practise HSC Mathematics Standard 2-style writing on area — multi-mark short answers and one structured extended response.

Master · Past-Paper Style

1. Short-answer questions

1.1 A trapezium has parallel sides 7 cm and 13 cm and a perpendicular height of 8 cm. Find its area. 2 marks Band 3

1.2 A circular fountain has a radius of 4.2 m. 3 marks Band 3-4
(a) Find its area in m² correct to 2 d.p.
(b) Convert this area to cm².

1.3 A composite shape is formed by removing a semicircle of radius 4 cm from a rectangle 14 cm × 10 cm. Find the area of the remaining shape correct to 2 d.p. 4 marks Band 4

Stuck on 1.3? Write the plan "Area = rectangle − semicircle" first to lock in subtraction; keep π exact until the very last line.

2. Extended response

2.1 A council is paving a circular plaza in the centre of a town. The plaza consists of a square paved area (20 m on each side) with a circular fountain (radius 3 m) in the exact centre. The square AND the circle are both paved with the same tiles.

The brief:

• Paving tiles cost $85 per m².

• The fountain itself (the water area) is NOT paved — only the square minus the circle.

(a) Calculate the area of the square paved region (without subtraction yet).
(b) Calculate the area of the circular fountain to 2 d.p.
(c) Calculate the area to be paved (square − circle) to 2 d.p., and the total paving cost. State your final answer in dollars and cents. 6 marks Band 5-6

Explicit marking criteria

Part (a) — 1 mark

• 1 mark — correct square area (20² = 400 m²).

Part (b) — 2 marks

• 1 mark — correct substitution into A = πr² using r = 3 (not d).

• 1 mark — correct evaluation to 2 d.p. using the π button.

Part (c) — 3 marks

• 1 mark — writes plan "Area = square − circle" (subtraction explicit).

• 1 mark — correct paved area to 2 d.p.

• 1 mark — correct total cost stated in dollars and cents with units.

Your response:

Stuck on (c)? The plan sentence "Area = square − circle" earns the first mark by itself, even before any numbers. Write it down first.

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What I'll revisit before next class:

Answers — sample responses + marking notes

1.1 — Trapezium (2 marks)

Sample response.
A = ½(a + b)h = ½(7 + 13) × 8 = ½ × 20 × 8 = 80 cm².

Marking notes. 1 mark — correct substitution (formula written + values placed). 1 mark — correct numerical answer with unit. Common error: forgetting the ½ gives 160 cm².

1.2 — Circular fountain (3 marks)

Sample response.
(a) A = πr² = π × (4.2)² = π × 17.64 = 55.4177... ≈ 55.42 m² (to 2 d.p.).
(b) 55.42 × 10 000 = 554 200 cm².

Marking notes. (a) 1 mark — correct substitution using π button. 1 mark — correct rounded answer. (b) 1 mark — correct conversion to cm² (× 10 000, NOT × 100). Common error: using × 100, giving 5542 cm² — wrong by a factor of 100.

1.3 — Rectangle − semicircle (4 marks)

Sample response.
Plan: Area = rectangle − semicircle.
Rectangle = 14 × 10 = 140 cm².
Semicircle (r = 4): A_semi = ½ × π × 16 = 8π = 25.1327... cm².
Remaining = 140 − 25.13 = 114.87 cm² (to 2 d.p.).

Marking notes. 1 mark — writes the subtraction plan. 1 mark — correct rectangle area. 1 mark — correct semicircle area using ½πr² with π exact. 1 mark — correct final answer with unit. Common error: adding instead of subtracting → 165.13 cm² (loses 2 marks).

2.1 — Paved plaza with fountain (6 marks): sample Band-6 response with annotations

Sample Band-6 response.

(a) Square area.

A_square = 20² = 400 m². [1 mark — square area.]

(b) Fountain area.

A_circle = πr² = π × 3² = 9π. [1 mark — substitution into πr² with r = 3.]
A_circle = 28.2743... ≈ 28.27 m² (to 2 d.p.). [1 mark — evaluation to 2 d.p.]

(c) Paved area and cost.

Plan: Paved area = square − circle. [1 mark — explicit subtraction plan.]
Paved = 400 − 28.27 = 371.73 m² (to 2 d.p.). [1 mark — correct paved area.]
Cost = 371.73 × $85 = $31,597.05. [1 mark — total cost in dollars and cents with units.]

Total: 6/6.

Band descriptors for marker.

Band 3: Square area correct; fountain area attempted but with d = 3 substituted into r² (giving A = π × 9 with wrong reasoning, accidentally correct), or 3.14 used instead of π. ≈ 2-3 marks.

Band 4: Both areas correct; adds instead of subtracts (giving 428.27 m²) OR forgets to write the subtraction plan. ≈ 4 marks.

Band 5: Full numerical solution including correct subtraction, but cost stated as bare "$31597" without cents, or units omitted. ≈ 5 marks.

Band 6: Complete, plan explicit, paved area rounded to 2 d.p., cost in dollars and cents with units. 6/6.