EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Prisms
Area and volume · Lesson 3 of 7
Last Lesson and Before
Answer each one, then check.
1. Last lesson: how many cm³ are in 1 litre?
1000
2. Area of a triangle with base 6 cm and height 8 cm?
24 cm²
3. How many faces does a cuboid have?
6
4. Work out 2(20 + 15 + 12).
94
Learning Objectives
1. Recognise prisms and their cross-sections.
2. Work out the volume of a cuboid and any prism.
3. Work out the surface area of a cuboid and a prism.
4. Solve problems involving volume and capacity.
What Makes a Prism
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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.
▸ Cuboid. V = l × w × h.
▸ Any prism. V = area of cross-section × length.
▸ Find the cross-section first. It may be a triangle, a trapezium or a compound shape - use last lesson's formulae.
▸ Units. cm³ or m³. For capacity, 1000 cm³ = 1 litre.
Volume of a Triangular Prism
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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. |
1. Area of the cross-section
½ × 6 × 8 = 24 cm²
2. Multiply by the length
24 × 12 = 288
Answer: 288 cm³
Surface Area
The surface area is the total area of all the faces. Sketching the net helps you not miss one.
▸ Cuboid. Three pairs of equal rectangles: SA = 2(lw + lh + wh).
▸ Triangular prism. Two triangles plus three rectangles.
▸ Rectangles round the side. Their total area is the perimeter of the cross-section times the length.
▸ Units. Surface area is an area: cm² or m².
Surface Area of a Triangular Prism
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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. |
1. Two triangles
2 × 24 = 48
2. Three rectangles: perimeter of the triangle × length
(6 + 8 + 10) × 12 = 288
3. Add
48 + 288 = 336
Answer: 336 cm²
How Long to Fill?
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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? |
1. Same units: cm
120 × 50 × 40 = 240 000 cm³
2. In litres
240 000 ÷ 1000 = 240 litres
3. Time
240 ÷ 4 = 60 minutes
Answer: 60 minutes (1 hour)
Key Terms
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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. |
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Volume The space inside a 3D shape, in cubic units. |
Surface area The total area of all the faces of a 3D shape. |
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Net A 2D shape that folds to make a 3D solid. |
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Your Task: Best Box
15 minutes
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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 = 10/3, about 593 cm³.
Can I...?
☐ Recognise a prism and its cross-section.
☐ Find the volume of a cuboid.
☐ Find the volume of any prism.
☐ Find the surface area of a cuboid.
☐ Find the surface area of a prism.
☐ Solve problems with volume and capacity.
Summary
✓ Volume of a prism = area of cross-section × length.
✓ Surface area = total area of all the faces.
✓ Convert to one unit before multiplying.
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EXAM FOCUS The diagram shows a triangular prism. Work out (a) its volume and (b) its surface area. (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. |