3
Surface area of prisms and pyramids from their nets
Discover Mathematics P6: Term 2, Unit 13, Lessons 1–2
GoalFind the surface area of a triangular prism and a square pyramid from their nets, and see what doubling does.
New words
prism · a solid with two equal parallel ends and rectangular sidespyramid · a solid with a base and triangular faces that meet at one pointslant height · the height of a triangular face of a pyramid, measured along the face
Explanation
For a solid with flat faces, unfold it into a net and add up the area of every face. A triangular prism has 2 triangles and 3 rectangles. For a triangle with sides 3, 4 and 5 cm (legs 3 and 4) and a prism length of 10 cm: each triangle is 3 × 4 ÷ 2 = 6 cm²; the rectangles are 3 × 10, 4 × 10 and 5 × 10. Total: 12 + 30 + 40 + 50 = 132 cm². A square pyramid has 1 square and 4 triangles. For a base of 6 cm and a slant height of 5 cm: the base is 36 cm², each triangle is 6 × 5 ÷ 2 = 15 cm², so the total is 36 + 4 × 15 = 96 cm². The slant height is always given.
| Solid | Faces | Surface area |
|---|---|---|
| Triangular prism | 2 triangles + 3 rectangles | 12 + 30 + 40 + 50 = 132 cm² |
| Square pyramid | 1 square + 4 triangles | 36 + 4 × 15 = 96 cm² |
Worked examples
The triangular prism: triangle legs 3 cm and 4 cm (long side 5 cm), prism length 10 cm. Triangles: 2 × 6 = 12 cm². Rectangles: 30 + 40 + 50 = 120 cm². Surface area = 132 cm².
The square pyramid: base 6 cm, slant height 5 cm. Base: 36 cm². Four triangles: 4 × 15 = 60 cm². Surface area = 96 cm².
Doubling. A cube has edge 2 cm: surface area 6 × 4 = 24 cm². Double the edge to 4 cm: 6 × 16 = 96 cm². The surface area is 96 ÷ 24 = 4 times bigger, not 2 times. The cuboid 5 by 3 by 2 (62 cm²) doubled to 10 by 6 by 4 has 248 cm², also 4 times bigger.
Class activity
Build a pyramid. Draw a net of a square pyramid on card: a square and four triangles. Cut, fold and tape it. Measure one triangle and compute the surface area.
Practice
1
A triangular prism has triangle legs 6 cm and 8 cm, long side 10 cm, and length 5 cm. Find its surface area.
Triangles: 2 × 24 = 48. Rectangles: 30 + 40 + 50 = 120. Total 168 cm².
2
A square pyramid has base 8 cm and slant height 5 cm. Find its surface area.
Base 64. Each triangle 8 × 5 ÷ 2 = 20. Total 64 + 80 = 144 cm².
3
A square pyramid has base 10 cm and slant height 13 cm. Find its surface area.
Base 100. Each triangle 10 × 13 ÷ 2 = 65. Total 100 + 260 = 360 cm².
4
A cube has edge 3 cm. Double the edge. Find both surface areas and how many times bigger the second is.
6 × 9 = 54 cm² and 6 × 36 = 216 cm². 216 ÷ 54 = 4 times.
5
How many faces does a square pyramid have in all, and what shapes are they?
5 faces: 1 square and 4 triangles.