Mathematics • Year 10 • Unit 1 • Lesson 16
Surface Area of Prisms — Skill Drill
Build fluency with the two prism surface-area methods from Lesson 16: the rectangular-prism formula SA = 2(lw + lh + wh), and the triangular-prism method "two triangles + three rectangles". One step at a time, from a fully worked example through guided practice to independent 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. A closed rectangular box has dimensions 12 cm by 8 cm by 5 cm. Calculate its total surface area.
Step 1 — Spot the rule.
Closed rectangular prism → use SA = 2(lw + lh + wh).
Reason: a closed prism has 6 faces. The formula counts each pair (top/bottom, front/back, left/right) only once.
Step 2 — Substitute l = 12, w = 8, h = 5.
SA = 2(12 × 8 + 12 × 5 + 8 × 5)
Reason: substitute the three dimensions into the three products inside the bracket.
Step 3 — Evaluate each product.
SA = 2(96 + 60 + 40)
Reason: 12 × 8 = 96, 12 × 5 = 60, 8 × 5 = 40.
Step 4 — Add inside the bracket, then double.
SA = 2 × 196 = 392
Reason: 96 + 60 + 40 = 196. Multiply by 2 because each face appears twice.
Step 5 — Write the answer with units.
SA = 392 cm²
Reason: surface area is measured in square units. Lengths were in cm, so the answer is in cm².
Answer: Total surface area = 392 cm².
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 triangular prism has a right-angled triangular base with sides 3 cm, 4 cm and 5 cm (5 cm is the hypotenuse). The prism is 10 cm long. Find the total surface area.
Step 1 — Spot the rule: a triangular prism has __________________ triangles and __________________ rectangles.
Step 2 — Area of one triangle (right-angled, legs 3 and 4):
A = ½ × 3 × 4 = ______ cm²
Step 3 — Two triangles together:
2 triangles = 2 × ______ = ______ cm²
Step 4 — Three rectangles (each side × prism length 10):
3 × 10 + 4 × 10 + 5 × 10 = ______ + ______ + ______ = ______ cm²
Step 5 — Add the two parts:
Total SA = ______ + ______ = ______ cm²
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (single shape, closed). The middle two are standard (open prism or triangular). The last two are extension (multi-step in syllabus).
Foundation — closed rectangular prisms
3.1 A closed rectangular prism has dimensions 5 cm by 4 cm by 3 cm. Find the surface area. 1 mark
3.2 A closed rectangular prism has dimensions 15 cm by 10 cm by 6 cm. Find the surface area. 1 mark
3.3 A cube has side length 7 cm. Find its surface area. 1 mark
3.4 A closed shoebox is 30 cm by 20 cm by 10 cm. Find the surface area. 1 mark
Standard — open prisms and triangular prisms
3.5 An open-top box has dimensions 10 cm by 8 cm by 6 cm. Find the surface area of the cardboard. 2 marks
3.6 A triangular prism has a base triangle with base 6 cm and perpendicular height 4 cm (the third side of the triangle is 5 cm). The prism length is 12 cm. Find the total surface area. 2 marks
Extension — push your thinking
3.7 An open-top fish tank measures 80 cm by 50 cm by 40 cm. Calculate the area of glass needed in cm² and in m². 3 marks
3.8 A cube has surface area 96 cm². A friend says "so the side length must be 96 ÷ 6 = 16 cm". Spot the friend's mistake and find the correct side length. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (3-4-5 triangular prism, 10 cm long)
Step 1: 2 triangles and 3 rectangles.
Step 2: A = ½ × 3 × 4 = 6 cm².
Step 3: 2 triangles = 2 × 6 = 12 cm².
Step 4: 3 × 10 + 4 × 10 + 5 × 10 = 30 + 40 + 50 = 120 cm².
Step 5: Total SA = 12 + 120 = 132 cm².
3.1 — 5 × 4 × 3 closed prism
SA = 2(5×4 + 5×3 + 4×3) = 2(20 + 15 + 12) = 2 × 47 = 94 cm².
3.2 — 15 × 10 × 6 closed prism
SA = 2(15×10 + 15×6 + 10×6) = 2(150 + 90 + 60) = 2 × 300 = 600 cm².
3.3 — Cube of side 7 cm
A cube has 6 identical square faces. SA = 6 × 7² = 6 × 49 = 294 cm².
3.4 — Shoebox 30 × 20 × 10
SA = 2(30×20 + 30×10 + 20×10) = 2(600 + 300 + 200) = 2 × 1100 = 2200 cm².
3.5 — Open-top 10 × 8 × 6 box
Closed SA = 2(10×8 + 10×6 + 8×6) = 2(80 + 60 + 48) = 376 cm².
Subtract the missing top = 10 × 8 = 80 cm².
Open SA = 376 − 80 = 296 cm².
3.6 — Triangular prism (3-4-5 triangle scaled to 6-4 with hyp 5? No — base 6, ht 4, third side 5)
Two triangles = 2 × ½ × 6 × 4 = 24 cm².
Three rectangles = 6 × 12 + 4 × 12 + 5 × 12 = 72 + 48 + 60 = 180 cm².
Total SA = 24 + 180 = 204 cm².
Use the perpendicular height (4) for the triangle area, not the third side (5).
3.7 — Open-top fish tank 80 × 50 × 40
Closed SA = 2(80×50 + 80×40 + 50×40) = 2(4000 + 3200 + 2000) = 18 400 cm².
Subtract the missing top = 80 × 50 = 4000 cm².
Glass needed = 18 400 − 4000 = 14 400 cm² = 1.44 m².
Convert: 10 000 cm² = 1 m², so 14 400 ÷ 10 000 = 1.44 m².
3.8 — Cube with SA = 96 cm²
The friend has divided by 6 to get 16 cm — but 16 is one face area, not a side length. A cube has 6 square faces of area side², so:
One face area = 96 ÷ 6 = 16 cm².
Side length = √16 = 4 cm.
Check: 6 × 4² = 6 × 16 = 96 ✓.