Mathematics Standard • Year 11 • Module 2 • Lesson 7
Surface Area of Prisms and Cylinders
Apply surface area to realistic Australian contexts: gift wrapping, water tanks, painting, and packaging.
Problem 1 — Wrapping a gift box
A rectangular gift box has dimensions 32 cm long, 22 cm wide, and 14 cm high. You want to estimate the amount of wrapping paper needed and add 10% for overlap.
Set up: What are we solving for?
(i) Calculate the total surface area of the gift box in cm². 2 marks
(ii) Add 10% extra for overlap. Find the amount of paper required to the nearest cm². 1 mark
(iii) Wrapping paper is sold in sheets of 50 cm × 70 cm. State whether one sheet is enough. Justify with a numerical comparison. 2 marks
Stuck? Revisit lesson § Card 3 — Rectangular Prism. List all six faces before adding.Problem 2 — Sheet metal for a rainwater tank
A backyard rainwater tank is cylindrical, open at the top. Its diameter is 2.4 m and its height is 1.8 m.
Set up: What are we solving for?
(i) Find the radius of the tank in metres. 1 mark
(ii) Calculate the area of sheet metal required (curved surface + base) in exact form (in terms of π) and to 2 d.p. 3 marks
(iii) Sheet metal costs $38 per m². Find the cost of materials to the nearest dollar. 1 mark
Stuck? Revisit lesson § Worked Example 4 — Open Cylinder. Open top means one circle + the curved surface, not two circles.Problem 3 — Tent fabric (triangular prism, no floor)
A camping tent is shaped like a triangular prism. The triangular cross-section is isosceles with base 2.4 m and slant sides 1.5 m each (the tent's two sloping side panels). The vertical height of the triangle (from the centre of the base up to the apex) is 0.9 m. The tent is 3.0 m long. The tent has no floor.
Set up: What are we solving for?
(i) Calculate the area of one triangular end (front or back) of the tent. 1 mark
(ii) Calculate the area of each of the three rectangular faces: two sloping side panels (each 1.5 m × 3.0 m) and the missing floor (2.4 m × 3.0 m). State which face is not included in the fabric. 2 marks
(iii) Calculate the total fabric area needed (two triangular ends + two sloping panels — floor omitted). 2 marks
Stuck? Revisit lesson § Card 4 — Triangular Prism. Lateral SA = perimeter of cross-section × length, but here you only include the two sloping rectangles.Problem 4 — Painting the outside of a garden shed
A timber garden shed is a rectangular prism: 3.0 m long, 2.0 m wide, and 2.2 m high. The shed has a single door that measures 0.8 m wide × 1.9 m high (the door is not being painted). Only the four external walls and the roof (top) are being painted — not the floor.
Set up: What are we solving for?
(i) Calculate the area of each of the four walls and the roof. State each face's area on a separate line. 2 marks
(ii) Subtract the door area. Find the total area to be painted in m². 1 mark
(iii) One litre of paint covers 12 m². Find the number of full 1-litre tins required (you cannot buy part-tins). Show your reasoning in one sentence. 2 marks
Stuck? Revisit lesson § Card 3 callout — list every face, then subtract the door area, then divide by coverage. Round UP for tins.Problem 5 — Label area for a soup can
A cylindrical soup can has diameter 7.5 cm and height 11 cm. The label wraps fully around the curved surface only (it does not cover the top or the bottom). The printers add an extra 1 cm of overlap along the seam.
Set up: What are we solving for?
(i) Calculate the radius of the can. Then find the curved surface area only, to 2 d.p. 2 marks
(ii) The unrolled curved surface is a rectangle. State its two dimensions (in cm) before any overlap is added. (Hint: one dimension is the circumference; the other is the height.) 2 marks
(iii) Add 1 cm of overlap to the wrap-around dimension only. Find the final label area in cm² to 2 d.p., and state the rectangle's printed size. 3 marks
Stuck? Revisit lesson § The Cylinder Key Insight — the unrolled label is a rectangle with width = circumference 2πr and height = h.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Gift box wrapping
Set up. Total SA of the closed rectangular prism, plus 10% overlap, then compared to one sheet's area.
(i) SA = 2(32×22) + 2(32×14) + 2(22×14) = 2(704) + 2(448) + 2(308) = 1408 + 896 + 616 = 2920 cm².
(ii) +10%: 2920 × 1.10 = 3212 cm². Required ≈ 3212 cm².
(iii) One sheet = 50 × 70 = 3500 cm². Since 3500 > 3212, one sheet is enough. (Even though the shape may not fit awkwardly — the question only asks about total area.)
Problem 2 — Open-top rainwater tank
Set up. Open top ⇒ one circular base + curved surface only.
(i) r = 2.4 ÷ 2 = 1.2 m.
(ii) SA = πr² + 2πrh = π(1.44) + 2π(1.2)(1.8) = 1.44π + 4.32π = 5.76π m² ≈ 18.10 m².
(iii) Cost = 18.10 × $38 = $687.79 ≈ $688. (Using unrounded 5.76π × 38 ≈ $687.76 also acceptable; the rounded $688 reflects the nearest dollar.)
Problem 3 — Tent fabric (no floor)
Set up. Two triangular ends + two sloping panels. The 2.4 × 3.0 floor rectangle is omitted.
(i) One triangle: A = ½ × 2.4 × 0.9 = 1.08 m².
(ii) Each sloping panel: 1.5 × 3.0 = 4.5 m². Two panels = 9.0 m². Floor (1.5 × 3.0?): floor is 2.4 × 3.0 = 7.2 m² (NOT included).
(iii) Fabric = 2(1.08) + 2(4.5) = 2.16 + 9.0 = 11.16 m².
Problem 4 — Painting the shed
Set up. Four walls + roof − door area, then divide by coverage and round up for whole tins.
(i) Two long walls: 3.0 × 2.2 = 6.6 m² each → 13.2 m².
Two short walls: 2.0 × 2.2 = 4.4 m² each → 8.8 m².
Roof: 3.0 × 2.0 = 6.0 m².
Subtotal = 13.2 + 8.8 + 6.0 = 28.0 m².
(ii) Door = 0.8 × 1.9 = 1.52 m². Total painted area = 28.0 − 1.52 = 26.48 m².
(iii) Tins = 26.48 ÷ 12 = 2.206... Round UP to 3 tins. You round up because 2 tins (covering 24 m²) would leave 2.48 m² of the shed unpainted — you must buy a third tin to finish the job.
Problem 5 — Soup can label
Set up. Curved surface area unrolls to a rectangle of width = circumference, height = can's height.
(i) r = 7.5 ÷ 2 = 3.75 cm. Curved SA = 2πrh = 2π(3.75)(11) = 82.5π ≈ 259.18 cm².
(ii) Rectangle dimensions: width = circumference = 2π(3.75) = 7.5π ≈ 23.56 cm; height = 11 cm. So 23.56 cm × 11 cm.
(iii) With 1 cm overlap on the wrap-around dimension: width becomes 23.56 + 1 = 24.56 cm. Printed size = 24.56 cm × 11 cm. Area = 24.56 × 11 = 270.18 cm².