Module 7

3D shapes & Pythagoras

Quick-reference revision notes for parents.

7.1 Properties of 3D shapes

Shape	Faces	Edges	Vertices
Cube	6	12	8
Cuboid	6	12	8
Triangular prism	5	9	6
Square-based pyramid	5	8	5
Tetrahedron	4	6	4
Cylinder	3*	2	0
Sphere	1	0	0

*counting curved surfaces.

Euler's rule for shapes with flat faces: F + V − E = 2.

A net is a 2D layout that folds up into the 3D shape. A cube has 11 different nets. The key: every face of the 3D shape appears once in the net.

Test for a cube net

It must have 6 squares. Mentally fold it — opposite faces should not be next to each other in the net.

Volume:

V = length × width × height

Surface area (sum of all six rectangles):

SA = 2(lw + lh + wh)

Worked example — cuboid 5 × 3 × 4 cm

V = 5 × 3 × 4 = 60 cm³
SA = 2(5×3 + 5×4 + 3×4) = 2(15 + 20 + 12) = 2 × 47 = 94 cm²

Watch out — units

Volume uses cm³, area uses cm², length uses cm. Mixing them up loses easy marks.

For a right-angled triangle with legs a, b and hypotenuse c (the longest side, opposite the right angle):

a² + b² = c²

Add the squares of the two shorter sides, then take the square root.

Worked example — legs 3 and 4

c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5

Subtract the squares.

Worked example — hypotenuse 13, one leg 5

a² = 13² − 5² = 169 − 25 = 144
a = √144 = 12

Many problems hide a right-angled triangle. Look for ladders, distances, diagonals.

Worked example

A 5m ladder leans against a wall, with its base 1.4m from the wall. How high up the wall does it reach?

h² = 5² − 1.4² = 25 − 1.96 = 23.04
h = √23.04 = 4.8 m