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Pythagoras in 3D Solids

In this worksheet, students use Pythagoras' Theorem to calculate lengths in cuboids.

'Pythagoras in 3D Solids' worksheet

Key stage:  KS 4

Curriculum topic:  Geometry and Measures

Curriculum subtopic:  Apply Pythagoras' Theorem and Trigonometric Ratios to Find Angles and Lengths in 2D Figures

Difficulty level:  

down

Worksheet Overview

QUESTION 1 of 10

Pythagoras' Theorem can easily be used to find lengths in right-angled triangles in cuboids.

 

Example

Find the length of the base diagonal BD of this 3 cm x 4cm x 9 cm cuboid to 3 sig. figs.

Hence find the length of the cuboid's diagonal BH to 3 sig. figs.

 

Answer

Locate the relevant right-angled triangle ABD on the cuboid's base.

 

Using Pythagoras' Theorem,

BD2 = 32 + 92 = 9 + 81 = 90

BD = √90 = 9.49 cm

 

Next, we look at triangle HDB, keeping the distance BD as √90 rather than 9.49.  This is because, when we square √90 we get exactly 90.

Using Pythagoras' Theorem,

BH2 = 42 + (√90)2 = 16 + 90 = 106

BH = √106 = 10.3 cm

 

Alternative Method

An alternative method is to say that:

BH2 = HD2 + AD2 + AB2 = 42 + 32 + 92 = 16 + 9 + 81 = 106

BH = √106 = 10.3 cm

Find the length of the base diagonal BD of this 2 cm x 3 cm x 7 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

Find the length of the cuboid's diagonal BH of this 2 cm x 3 cm x 7 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

Find the length of the base diagonal BD of this 3 cm x 5 cm x 11 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

Find the length of the cuboid's diagonal BH of this 3 cm x 5 cm x 11 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

Find the length of the base diagonal BD of this 4 cm x 5 cm x 12 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

Find the length of the cuboid's diagonal BH of this 4 cm x 5 cm x 12 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

Find the length of the base diagonal AC of this 4 cm x 5 cm x 10 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

Find the length of the cuboid's diagonal AG of this 4 cm x 5 cm x 10 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

Find the length of the base diagonal AC of this 5 cm x 5.5 cm x 14 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

Find the length of the cuboid's diagonal AG of this 5 cm x 5.5 cm x 14 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

  • Question 1

Find the length of the base diagonal BD of this 2 cm x 3 cm x 7 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

CORRECT ANSWER
7.28
EDDIE SAYS
Use Pythagoras' Theorem to get:
BD² = 2² + 7² = 4 + 49 = 53
  • Question 2

Find the length of the cuboid's diagonal BH of this 2 cm x 3 cm x 7 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

CORRECT ANSWER
7.87
EDDIE SAYS
Use Pythagoras' Theorem to get:
BH² = 2² + 7² + 3² = 4 + 49 + 9 = 62
  • Question 3

Find the length of the base diagonal BD of this 3 cm x 5 cm x 11 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

CORRECT ANSWER
11.4
EDDIE SAYS
Use Pythagoras' Theorem to get:
BD² = 3² + 11² = 9 + 121 = 130
  • Question 4

Find the length of the cuboid's diagonal BH of this 3 cm x 5 cm x 11 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

CORRECT ANSWER
12.4
EDDIE SAYS
Use Pythagoras' Theorem to get:
BH² = 3² + 5² + 11² = 9 + 25 + 121 = 155
  • Question 5

Find the length of the base diagonal BD of this 4 cm x 5 cm x 12 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

CORRECT ANSWER
12.6
EDDIE SAYS
Use Pythagoras' Theorem to get:
BD² = 12² + 4² = 144 + 16 = 160
  • Question 6

Find the length of the cuboid's diagonal BH of this 4 cm x 5 cm x 12 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

CORRECT ANSWER
13.6
EDDIE SAYS
Use Pythagoras' Theorem to get:
BH² = 4² + 5² + 12² = 16 + 25 + 144 = 185
  • Question 7

Find the length of the base diagonal AC of this 4 cm x 5 cm x 10 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

CORRECT ANSWER
10.8
EDDIE SAYS
Use Pythagoras' Theorem to get:
AC² = 4² + 10² = 16 + 100 = 116
  • Question 8

Find the length of the cuboid's diagonal AG of this 4 cm x 5 cm x 10 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

CORRECT ANSWER
11.9
EDDIE SAYS
Use Pythagoras' Theorem to get:
AG² = 4² + 5² + 10² = 16 + 25 + 100 = 141
  • Question 9

Find the length of the base diagonal AC of this 5 cm x 5.5 cm x 14 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

CORRECT ANSWER
14.9
EDDIE SAYS
Use Pythagoras' Theorem to get:
AC² = 5² + 14² = 25 + 196 = 221
  • Question 10

Find the length of the cuboid's diagonal AG of this 5 cm x 5.5 cm x 14 cm cuboid in cm to 3 sig. figs.

(Just write the number)

 

CORRECT ANSWER
15.9
EDDIE SAYS
Use Pythagoras' Theorem to get:
AG² = 5² + 5.5² + 14² = 25 + 30.25 + 196 = 251.25
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