Pythagoras in 3D

The Maths Habit

  1. Find x.

    x = 29 mm

  2. Find x to 1 decimal place.

    x = 11.5 m

  3. Find x.

    x = 6 cm

You want to fit the longest straight stick you can inside a shoebox. Where does it go?

From a bottom corner to the opposite top corner

Find the square root, then square it.

  1. √49

    7, then 49

  2. √30

    5.477…, then 30

  3. √72

    8.485…, then 72

  4. (√58)2

    58

A cuboid is 6 cm by 6 cm by 7 cm. Find its diagonal.

6 cm6 cm7 cm6 cm6 cmbase2​=62​ + 62​=72√72keep it as 72√727 cmdiagonal2​=72 + 72​=121diagonal=√121 = 11 cm11 cm

Find the length of the diagonal of the cuboid.

6 cm6 cm7 cm
base2 =62 + 62base2 =72diagonal2 =72 + 72diagonal2 =121diagonal =√121 = 11 cm
6 cm6 cmbasebase7 cmdiagonal6 cm6 cm7 cm

Find the length of the diagonal of the cuboid.

8 cm9 cm12 cm
base2 =82 + 92base2 =145diagonal2 =145 + 122diagonal2 =289diagonal =√289 = 17 cm
8 cm9 cmbasebase12 cmdiagonal8 cm9 cm12 cm

Find the length of the diagonal of the cuboid.

6 cm6 cm7 cm
base2 =62 + 62base2 =72diagonal2 =72 + 72diagonal2 =121diagonal =√121 = 11 cm
6 cm6 cmbasebase6 cmdiagonal6 cm6 cm6 cm

Find the length of the diagonal of the cube to 1 decimal place.

6 cm6 cm6 cm
base2 =62 + 62base2 =72diagonal2 =72 + 62diagonal2 =108diagonal =√108 = 10.4 cm
6 cm6 cmbasebase6 cmdiagonal6 cm6 cm6 cm

Find the length of a sloping edge of the pyramid.

12 cm7 cm
half diagonal2 =62 + 62half diagonal2 =72edge2 =72 + 72edge2 =121edge =√121 = 11 cm
6 cm6 cmhalf diagonalhalf diagonal7 cmedge12 cm7 cm

Find the length of a sloping edge of the pyramid.

10 cm8 cm
half diagonal2 =52 + 52half diagonal2 =50edge2 =50 + 82edge2 =114edge =√114 = 10.7 cm
5 cm5 cmhalf diagonalhalf diagonal8 cmedge10 cm8 cm

Find the length of a sloping edge of the pyramid.

12 cm7 cm
half diagonal2 =62 + 62half diagonal2 =72edge2 =72 + 72edge2 =121edge =√121 = 11 cm
6 cm6 cmhalf diagonalhalf diagonal7 cmedge12 cm7 cm

Find the length of a sloping edge of the pyramid.

4 cm9 cm
half diagonal2 =22 + 22half diagonal2 =8edge2 =8 + 92edge2 =89edge =√89 = 9.4 cm
2 cm2 cmhalf diagonalhalf diagonal9 cmedge4 cm9 cm

Find the length of a sloping edge of the pyramid.

12 cm7 cm
half diagonal2 =62 + 62half diagonal2 =72edge2 =72 + 72edge2 =121edge =√121 = 11 cm
6 cm6 cmhalf diagonalhalf diagonal7 cmedge12 cm7 cm

Find the length of the diagonal of the cuboid.

6 cm3 cm2 cm
base2 =62 + 32base2 =45diagonal2 =45 + 22diagonal2 =49diagonal =√49 = 7 cm
6 cm3 cmbasebase2 cmdiagonal6 cm3 cm2 cm

Find the length of a sloping edge of the pyramid.

12 cm7 cm
half diagonal2 =62 + 62half diagonal2 =72edge2 =72 + 72edge2 =121edge =√121 = 11 cm
6 cm6 cmhalf diagonalhalf diagonal7 cmedge12 cm7 cm

Find the length of a sloping edge of the pyramid.

8 cm5 cm
half diagonal2 =42 + 42half diagonal2 =32edge2 =32 + 52edge2 =57edge =√57 = 7.5 cm
4 cm4 cmhalf diagonalhalf diagonal5 cmedge8 cm5 cm

Find the length of a sloping edge of the pyramid.

12 cm7 cm
half diagonal2 =62 + 62half diagonal2 =72edge2 =72 + 72edge2 =121edge =√121 = 11 cm
6 cm6 cmhalf diagonalhalf diagonal7 cmedge12 cm7 cm

Find the length of the diagonal of the cuboid to 1 decimal place.

5 cm8 cm10 cm
base2 =52 + 82base2 =89diagonal2 =89 + 102diagonal2 =189diagonal =√189 = 13.7 cm
5 cm8 cmbasebase10 cmdiagonal5 cm8 cm10 cm

Pythagoras in 3D worksheet, page 1

Pythagoras in 3D worksheet, page 2

Answer sheet

  1. A pyramid has a square base of side 8 cm. Each sloping edge is 9 cm. Find its height. 7 cm

  2. A pyramid has a square base of side 6 cm and a height of 4 cm. Find the length of a sloping edge. 5.8 cm

  3. Will a 56 cm umbrella fit in a box 50 cm by 20 cm by 15 cm? No: the diagonal is 55.9 cm

  1. Find the length of the diagonal shown in the cuboid.

    7 cm
    6 cm2 cm3 cm

  2. A cube has edges of 5 cm. Find the length of its longest diagonal to 1 decimal place.

    8.7 cm

  3. A box is 30 cm by 25 cm by 12 cm. Will a 40 cm rod fit inside it? Show how you know.

    Yes: 40.9 cm
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