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GCSE Maths · Higher · Chapter 11: Right-angled triangles · Lesson 2

Pythagoras in three dimensions

Grades 5–8About 45 minutesCalculator allowed33 marks
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NameClassDate
A

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

  1. A1
    Needed today

    A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse.

    [1 mark]
    Answer
  2. A2
    From chapter 1

    Work out .

    [1 mark]
    Answer
  3. A3
    From chapter 9

    A cone has radius 5 cm and height 6 cm. Find its volume in terms of .

    [1 mark]
    Answer
  4. A4
    Key Stage 3

    Work out .

    [1 mark]
    Answer
B

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it.

Example 1

A cuboid is 24 cm by 10 cm by 20 cm. Work out the longest rod that fits inside it, to 1 decimal place.

24 cm20 cm10 cm
Not drawn accurately
Hint Find the diagonal of the base first, then use it with the height.
  1. B1

    A cuboid is 16 cm by 12 cm by 9 cm. Work out the longest rod that fits inside it, to 1 decimal place.

    [3 marks]
    16 cm9 cm12 cm
    Not drawn accurately
    Answer
Example 2

A pyramid has a square base of side 24 cm. Its apex is 16 cm above the centre of the base. Work out .

VMO24 cm
Not drawn accurately
Hint is the middle of an edge, so is half a side. has a right angle at .
  1. B2

    A pyramid has a square base of side 40 cm. Its apex is 21 cm above the centre of the base. Work out .

    [3 marks]
    VMO40 cm
    Not drawn accurately
    Answer
C

Practice

Grades 5–7
  1. C1

    A cuboid is 20 cm long and 15 cm wide. Its diagonal is 29 cm. Work out its height , to 1 decimal place.

    [3 marks]
    AG20 cmh15 cm
    Not drawn accurately
    Answer
  2. C2

    A pyramid has a square base of side 14 cm. Its apex is 9 cm above the centre of the base. Work out the sloping edge , to 1 decimal place.

    [3 marks]
    VQO14 cm
    Not drawn accurately
    Answer
  1. C3

    A cone has a base diameter of 18 cm and a vertical height of 12 cm. Work out its slant height.

    [2 marks]
    18 cm12 cm
    Not drawn accurately
    Answer
  2. C4

    is the point and is the point . Work out the distance , to 1 decimal place.

    [3 marks]
    Answer
D

Exam-style questions

Grades 6–8
  1. D1

    A cuboid is 12 cm by 9 cm by 8 cm. Mia says, "The longest line inside it is cm."

    Explain why Mia is wrong, and work out the correct length.

    [3 marks]
    12 cm8 cm9 cm
    Not drawn accurately
  1. D2

    A tent is a pyramid with a square base of side 3.2 m. Each of its four sloping edges is 2.9 m long. The top of the tent is directly above the centre of the base.

    V3.2 m2.9 mh
    Not drawn accurately
    (a)

    Work out the height of the tent. Give your answer to 3 significant figures. You must show your working.

    [4 marks]
    Answer
    (b)

    Kai is 1.85 m tall. Can he stand up straight in the middle of the tent? Give a reason.

    [1 mark]
    Total for D2: 5 marks
E

Extension

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

  1. E1

    The edges of a cuboid are in the ratio . The longest straight line inside it is 24 cm. Work out the volume of the cuboid.

    [4 marks]
    Hint 1 · What to try

    Call the shortest edge cm. What are the other two?

    Hint 2 · The first line

    .

    Hint 3 · The full method

    , so and the edges are 8, 16 and 16 cm.

    Answer
F

How did it go?

Colour one circle on each line.
Not yetNearlyYes
I can use Pythagoras twice to find the longest line inside a cuboid.
I can find a sloping length inside a pyramid or a cone.
I can work backwards to find an edge or a height.

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