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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Got it Partly Lost

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.