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Maths · Area and volume
Prisms
A prism has the same cross-section all the way through. Its volume is the area of that cross-section times its length, and its surface area is the total area of all its faces.
Last Lesson and Before
Answer each one, then check.
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1
Last lesson: how many cm³ are in 1 litre?
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1000
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2
Area of a triangle with base 6 cm and height 8 cm?
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24 cm²
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3
How many faces does a cuboid have?
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6
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4
Work out \(2(20 + 15 + 12)\).
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94
Learning Objectives
- 1Recognise prisms and their cross-sections.
- 2Work out the volume of a cuboid and any prism.
- 3Work out the surface area of a cuboid and a prism.
- 4Solve problems involving volume and capacity.
What Makes a Prism
Slice a prism anywhere at right angles to its length and you get the same shape - the cross-section. So its volume is the area of the cross-section multiplied by the length. A cuboid is just a prism with a rectangular cross-section.
The cross-section is the same all the way along.
Volume
Volume is the space inside a 3D shape, measured in cubic units.
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Cuboid
\(V = l \times w \times h\).
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Any prism
\(V = \text{area of cross-section} \times \text{length}\).
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Find the cross-section first
It may be a triangle, a trapezium or a compound shape - use last lesson's formulae.
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Units
cm³ or m³. For capacity, 1000 cm³ = 1 litre.
Volume of a Triangular Prism
A prism has a right-angled triangle as its cross-section, with shorter sides 6 cm and 8 cm. The prism is 12 cm long. Work out its volume.
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- 1 Area of the cross-section \(\frac{1}{2} \times 6 \times 8 = 24\) cm²
- 2 Multiply by the length \(24 \times 12 = 288\)
Answer288 cm³
Surface Area
The surface area is the total area of all the faces. Sketching the net helps you not miss one.
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Cuboid
Three pairs of equal rectangles: \(SA = 2(lw + lh + wh)\).
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Triangular prism
Two triangles plus three rectangles.
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Rectangles round the side
Their total area is the perimeter of the cross-section times the length.
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Units
Surface area is an area: cm² or m².
Surface Area of a Triangular Prism
Work out the surface area of the prism above. Its triangle has sides 6 cm, 8 cm and 10 cm, and it is 12 cm long.
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- 1 Two triangles \(2 \times 24 = 48\)
- 2 Three rectangles: perimeter of the triangle × length \((6 + 8 + 10) \times 12 = 288\)
- 3 Add \(48 + 288 = 336\)
Answer336 cm²
How Long to Fill?
A tank is a cuboid 1.2 m long, 50 cm wide and 40 cm deep. Water flows in at 4 litres per minute. How long does it take to fill?
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- 1 Same units: cm \(120 \times 50 \times 40 = 240\,000\) cm³
- 2 In litres \(240\,000 \div 1000 = 240\) litres
- 3 Time \(240 \div 4 = 60\) minutes
Answer60 minutes (1 hour)
Best Box
A box with no lid is made from a 20 cm by 20 cm square of card by cutting a square of side \(x\) cm from each corner and folding up the sides. Work out the volume for \(x = 1, 2, 3, 4, 5\) and 6. Which value of \(x\) gives the biggest box?
1. Write the length and width in terms of \(x\).
2. Work out each volume.
3. Look for the maximum.
A good answer shows: \(V = x(20 - 2x)^2\): 324, 512, 588, 576, 500, 384 cm³. The biggest of these is at \(x = 3\) (588 cm³); the true maximum is at \(x = \frac{10}{3}\), about 593 cm³.
Can I...?
- 1Recognise a prism and its cross-section.
- 2Find the volume of a cuboid.
- 3Find the volume of any prism.
- 4Find the surface area of a cuboid.
- 5Find the surface area of a prism.
- 6Solve problems with volume and capacity.
Summary & Exam Focus
- Volume of a prism = area of cross-section × length.
- Surface area = total area of all the faces.
- Convert to one unit before multiplying.
Exam focus
The diagram shows a triangular prism. Work out (a) its volume and (b) its surface area. (5 marks) (5 marks)
For surface area, list the faces before you calculate - "2 triangles, rectangles 6 × 12, 8 × 12 and 10 × 12" - so you can see you have them all.
Key terms
The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.
- Prism
- A 3D shape with the same cross-section all along its length.
- Cross-section
- The shape you get when you slice through a solid at right angles to its length.
- Volume
- The space inside a 3D shape, in cubic units.
- Surface area
- The total area of all the faces of a 3D shape.
- Net
- A 2D shape that folds to make a 3D solid.
Practice questions
Have a go at each one before you open its answer.
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Question 1 Non-calculator 2 marks
A cuboid is 5 cm long, 4 cm wide and 3 cm high. Work out (a) its volume and (b) its surface area.
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Model answer
(a) \(5 \times 4 \times 3 = 60\) cm³ (b) \(2(20 + 15 + 12) = 94\) cm²
Mark scheme
- (a) 60 cm³ — B1
- (b) 94 cm² — B1
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Question 2 Non-calculator 5 marks
The diagram shows a triangular prism. The cross-section is a right-angled triangle with sides 6 cm, 8 cm and 10 cm. The prism is 12 cm long. Work out (a) the volume and (b) the total surface area of the prism.
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Model answer
(a) \(\frac{1}{2} \times 6 \times 8 \times 12 = 288\) cm³ (b) \(2 \times 24 + 6 \times 12 + 8 \times 12 + 10 \times 12 = 48 + 72 + 96 + 120 = 336\) cm²
Mark scheme
- (a) \(\frac{1}{2} \times 6 \times 8\) — M1
- (a) 288 cm³ — A1
- (b) Area of 2 triangles, 48 — M1
- (b) At least two of 72, 96, 120 — M1
- (b) 336 cm² — A1
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Question 3 Calculator 3 marks
A tank is a cuboid 1.2 m long, 50 cm wide and 40 cm deep. It is empty. Water flows in at 4 litres per minute. How long does it take to fill the tank?
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Model answer
\(120 \times 50 \times 40 = 240\,000\) cm³ = 240 litres. \(240 \div 4 = 60\) minutes.
Mark scheme
- Volume with consistent units — P1
- 240 litres — P1
- 60 minutes — A1
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Question 4 Non-calculator 3 marks
A prism has a cross-section in the shape of a trapezium, with parallel sides 4 cm and 6 cm and height 3 cm. The volume of the prism is 225 cm³. Work out the length of the prism.
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Model answer
Area of cross-section \(\frac{1}{2}(4 + 6) \times 3 = 15\) cm². Length \(225 \div 15 = 15\) cm.
Mark scheme
- 15 cm² — M1
- \(225 \div 15\) — M1
- 15 cm — A1
Quick check
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What is the volume of a cube with side 4 cm?
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B: 64 cm³
\(4 \times 4 \times 4 = 64\).
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A prism has cross-section area 18 cm² and length 10 cm. What is its volume?
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A: 180 cm³
Volume = area of cross-section × length = \(18 \times 10\).
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What is the surface area of a cube with side 5 cm?
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D: 150 cm²
6 square faces, each \(5 \times 5 = 25\): \(6 \times 25 = 150\).
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