Year 11 Standard · Unit 2 Right-angled triangles and circle geometry · Lesson 3 of 8

Finding the flat triangle inside a solid

A box, a room and a pyramid all hide right-angled triangles. This lesson finds them, one at a time, to get a length or an angle that no single triangle gives.

Geometry and TrigonometryGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: a right-angled triangle has short sides 5 cm and 12 cm. Find the hypotenuse.
    Answer13 cm, because 25 + 144 = 169 and √169 = 13.
  2. From Year 10: a cone has radius 6 cm and vertical height 8 cm. Find its slant height.
    Answer10 cm, because 36 + 64 = 100.
  3. From Year 11: a ramp rises 2 m over a horizontal distance of 10 m. Which angle of elevation does it make, to the nearest degree?
    Answer11°, because tan−1(2 ÷ 10) ≈ 11.3°.
Words used in this lesson
space diagonal
a line from one corner of a solid to the opposite corner, through the inside. For example: the longest rod that fits in a box.
cuboid
a solid with six rectangular faces. For example: a shoebox.
floor diagonal
the diagonal of the base of a solid. For example: the line across the floor of a room.
sloping edge
an edge of a pyramid from the apex to a corner of the base. For example: the edge from the top of a pyramid-shaped tent to a corner peg.
projection
the shadow of a line on a plane, straight below it. For example: half the base diagonal, under a sloping edge of a pyramid.
apex
the top point of a pyramid. For example: the tip of a roof.

A ruler in a box

A box is 20 cm long, 15 cm wide and 12 cm high. Mei Lin has a 30 cm ruler and wants to lay it inside the box, at any angle she likes, without bending it.

Predict first: does the ruler fit? Write down yes or no and a reason, then use the first figure to see which triangles decide it.

4
3
2
5floor diagonal
√29 ≈ 5.39space diagonal

A 4 by 3 by 2 box: floor diagonal √(4² + 3²) = 5, space diagonal √(25 + 4) = √29 ≈ 5.39. The sum of the edges is 9.

Show me the steps
  1. A 4 by 3 by 2 box: floor diagonal √(16 + 9) = 5, space diagonal √(25 + 4) = √29 ≈ 5.39. The sum of the edges, 9, is much more.
  2. A 2 by 3 by 6 box: floor diagonal √13 ≈ 3.61, and the space diagonal √(13 + 36) = 7 exactly, a whole number.
  3. A cube 8 by 8 by 8: floor diagonal √128 ≈ 11.31, space diagonal √192 ≈ 13.86, which is 8√3.
  4. A 5 by 5 by 2 box: space diagonal √54 ≈ 7.35. It is longer than any edge and shorter than the sum, 12, in every setting.

Use Pythagoras twice. The floor diagonal is √(l² + w²). That diagonal and the height h are the legs of a second right-angled triangle, and its hypotenuse is the space diagonal, √(l² + w² + h²).

An angle between a line and a plane is the angle between the line and its shadow on the plane, found in the right-angled triangle made by the line, its shadow and the vertical. For a square-based pyramid the shadow of an edge is half the base diagonal.

8
6
√68 ≈ 8.25sloping edge
46.7°angle of the edge with the base

Base edge 8, height 6: half diagonal ≈ 5.66, sloping edge √(6² + 5.66²) = √68 ≈ 8.25, angle with the base 46.7°.

Show me the steps
  1. Base edge 8, height 6: half the diagonal is 8√2 ÷ 2 ≈ 5.66, the edge is √(36 + 32) = √68 ≈ 8.25, and the angle with the base is tan⁻¹(6 ÷ 5.66) ≈ 46.7°.
  2. Base edge 4, height 3 (everything halved): the edge is √17 ≈ 4.12 and the angle is still 46.7°. A smaller copy keeps its angles.
  3. Base edge 12, height 3: a flat pyramid. Half the diagonal is about 8.49, the edge is 9 exactly, and the angle falls to 19.5°.
  4. Base edge 6, height 12: a tall spike. The edge is √162 ≈ 12.73 and the angle rises to 70.5°. A taller pyramid makes steeper edges.

The longest line in a box is not the sum of its edges. A box 1 by 2 by 2 has a space diagonal of √9 = 3, not 1 + 2 + 2 = 5. And half of a pyramid's base edge is the wrong leg for a sloping edge: use half the diagonal, which is longer than half the edge.

Worked examples

1. Does a 30 cm ruler fit in a box 20 cm by 15 cm by 12 cm?

  1. Floor diagonal: √(20² + 15²) = √625 = 25 cm.
  2. Space diagonal: √(25² + 12²) = √769 ≈ 27.7 cm. The ruler is longer, so it does not fit.

2. A cuboid has a 20 cm by 21 cm base and is 10 cm high. Find the angle between its space diagonal and the base, to 1 decimal place.

  1. The shadow on the base is the floor diagonal: √(400 + 441) = 29 cm.
  2. tan θ = 10 ÷ 29, so θ = 19.0°.

3. A square-based pyramid has base edge 12 cm and vertical height 8 cm. Find a sloping edge and its angle with the base, to 1 decimal place.

  1. Half the base diagonal is 12√2 ÷ 2 ≈ 8.49 cm, so the edge is √(8² + 72) = √136 ≈ 11.7 cm.
  2. tan θ = 8 ÷ 8.49, so θ = 43.3°.

Your go. The first step is given. Find the space diagonal of a cuboid 7 cm by 24 cm by 60 cm.

  1. The floor diagonal is √(7² + 24²) = √625 = 25 cm.
  2. Now the floor diagonal and the height 60 are the legs of the next triangle. What is its hypotenuse?
Check the answer

√(25² + 60²) = √4225 = 65 cm. Adding the three edges would give 91, which is longer than the diagonal can ever be.

With a GDC (optional)

Keep the squares inside one square root: enter the root key, then the whole sum, then close the bracket and press equals. That way nothing is rounded in the middle. For an angle, use the inverse tangent key on the fraction, with brackets around the top and the bottom if either is a calculation.

Check the size by hand first: a space diagonal is longer than every edge, and shorter than the sum of the three. Everything here can be done with a scientific calculator.

Check before you go on

1. A cuboid is 3 cm by 4 cm by 12 cm. What is the length of its space diagonal?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A room is 9 m long and 12 m wide, and 8 m high. How long is the diagonal of the floor, in m?

3. For the same room (9 m by 12 m by 8 m), how long is the longest straight line from a floor corner to the opposite ceiling corner, in m?

Level 2 · Building

4. A cuboid is 6 cm by 7 cm by 9 cm. Find the length of its space diagonal, to 1 decimal place.

5. A cuboid has a base 8 cm by 6 cm and is 5 cm high. Find the angle between its space diagonal and the base, to 1 decimal place.

6. A square-based pyramid has base edge 14 cm and vertical height 12 cm. How long is a sloping edge, to 1 decimal place?

Level 3 · Stretch

7. A square-based pyramid has base edge 16 m and vertical height 6 m. Find the angle between a sloping edge and the base, to 1 decimal place.

8. A cuboid has a square base of side 6 cm. Its space diagonal is 11 cm long. How high is the cuboid, in cm?

Something to think about

A cube has edge a. Without a calculator, write its space diagonal in terms of a. Then say how long the diagonal is compared with the edge: more than double, or less?

One way to think about it

The floor diagonal is a√2 and the space diagonal is √(2a² + a²) = a√3 ≈ 1.73a. That is less than double the edge, and more than the floor diagonal a√2 ≈ 1.41a: each added dimension lengthens it a little less.

Exit ticket

9. A storeroom measures 1 m by 4 m by 8 m. What is the longest straight pole that fits inside it?