3
Review: area, surface area and volume
Discover Mathematics P6: Term 2, Units 8–13, Review
GoalReview the areas of parallelograms, triangles and trapeziums, surface area and volume.
New words
area · the flat space inside a shape, in square unitssurface area · the sum of the areas of all the faces of a solidvolume · the space inside a solid, in cubic units
Explanation
Areas: parallelogram = base × height; triangle = base × height ÷ 2; trapezium = (a + b) × height ÷ 2, where a and b are the parallel sides. The height is always perpendicular to the base. Surface area of a cuboid = 2 × (l × w + l × h + w × h); of a cube = 6 × a × a; for a prism or a pyramid add the faces of the net. Volume of a cuboid = l × w × h. When all dimensions double, the surface area is 4 times bigger and the volume 8 times bigger. Always write square units (cm²) for area and surface area, and cubic units (cm³) for volume.
| Measure | Rule | Unit |
|---|---|---|
| Parallelogram | b × h | cm² |
| Triangle | b × h ÷ 2 | cm² |
| Trapezium | (a + b) × h ÷ 2 | cm² |
| Cuboid surface | 2(lw + lh + wh) | cm² |
| Cuboid volume | l × w × h | cm³ |
Worked examples
Parallelogram, base 9 cm, height 6 cm: 54 cm². Triangle, base 10 cm, height 7 cm: 10 × 7 ÷ 2 = 35 cm². Trapezium, parallel sides 5 cm and 9 cm, height 6 cm: (5 + 9) × 6 ÷ 2 = 42 cm².
A cuboid is 6 cm by 4 cm by 3 cm. Surface area = 2 × (24 + 18 + 12) = 108 cm². Volume = 6 × 4 × 3 = 72 cm³.
A cube has edge 5 cm: surface area 6 × 25 = 150 cm², volume 5 × 5 × 5 = 125 cm³. Double the edge to 10 cm: 600 cm² (4 times) and 1,000 cm³ (8 times).
Class activity
Measure a room. With a grown-up, measure the length, width and height of a room or a large box. Find its volume, and the area of the floor. Write the right unit for each.
Practice
1
A triangle has area 36 cm² and base 9 cm. Find its height.
36 × 2 ÷ 9 = 8 cm.
2
A cuboid is 10 cm by 5 cm by 2 cm. Find its surface area and volume.
Surface area: 2 × (50 + 20 + 10) = 160 cm². Volume: 100 cm³.
3
How many boxes 4 cm by 3 cm by 2 cm fit in a box 12 cm by 9 cm by 6 cm?
3 × 3 × 3 = 27 boxes.
4
A square pyramid has base 4 cm and slant height 6 cm. Find its surface area.
Base 16. Each triangle 4 × 6 ÷ 2 = 12. Total 16 + 48 = 64 cm².
5
A trapezium has area 45 cm² and parallel sides 4 cm and 14 cm. Find its height.
45 × 2 ÷ 18 = 5 cm.