Year 11 Extended · Unit 1 Right-angled triangles and circle geometry · Lesson 2 of 7

Lines and planes in solids: the angle a line or a face makes with the ground

A supplier wants to slide a long bar into a container. This lesson finds the longest line that fits inside a box and the angle it makes with the floor, then the angle between a sloping face and the base, by finding the right-angled triangle that holds the angle.

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 legs 5 cm and 12 cm. Find the hypotenuse.
    Answer13 cm.
  2. From Year 10: a right-angled triangle has an opposite side of 6 and an adjacent side of 8. Find the angle, to 1 decimal place.
    Answer36.9°, from tan−1(6 ÷ 8).
  3. From Year 9: the diagonal of a rectangle 9 cm by 4 cm, to 1 decimal place.
    Answer9.8 cm, from √(81 + 16).
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 in a box.
plane
a flat surface that goes on in every direction. For example: a table top, thought of as endless.
perpendicular
at a right angle to. For example: a wall meets the floor at a perpendicular.
shadow of a line
the line you get by dropping each point of it straight onto a plane. For example: a stick's shadow at noon.
slant
the line up the middle of a sloping face, perpendicular to the base edge. For example: the line a ball would roll down a roof.

A bar in a container

A steel container is 12 m long, 5 m wide and 6 m high. A supplier wants to fit a 14 m bar inside it, straight and without bending it.

Predict first: the longest edge is only 12 m. Will the 14 m bar fit, or not? Give a reason before you use the first figure.

6
14.3 mlongest bar that fits
24.8°angle of the bar with the floor

Floor 12 by 5, height 6: floor diagonal 13.0, longest bar √(13² + 6²) ≈ 14.3 m, angle with the floor tan⁻¹(6 ÷ 13) ≈ 24.8°.

Show me the steps
  1. Floor 12 by 5, height 6: the floor diagonal is 13.0, the longest bar √(13² + 6²) ≈ 14.3 m and it rises at tan⁻¹(6 ÷ 13) ≈ 24.8°.
  2. Raise the height to 12: the bar is √(169 + 144) ≈ 17.7 m and the angle is tan⁻¹(12 ÷ 13) ≈ 42.7°. Doubling the height did not double the bar.
  3. Lower the height to 4: the bar is √(169 + 16) ≈ 13.6 m, barely longer than the floor diagonal, and the angle is tan⁻¹(4 ÷ 13) ≈ 17.1°.
  4. The longest bar is always longer than the floor diagonal and is the hypotenuse of a triangle that stands on it. The angle with the floor is that triangle's angle at the floor corner.

To find a length or an angle in a solid, find the right-angled triangle that holds it. The longest bar in a cuboid is the hypotenuse of a triangle that stands on the floor diagonal, so it is √(l² + w² + h²). The angle between a line and a plane is the angle between the line and its own shadow on the plane, found by dropping a perpendicular from a point of the line.

The angle between two planes is measured at right angles to the line where they meet: draw one line in each plane perpendicular to the join, at the same point. In a pyramid with its apex above the centre, the face's line runs to the midpoint of a base edge, so its horizontal leg is half a side.

8
6
56.3°angle of a face with the base
46.7°angle of an edge with the base

Base 8, height 6: face tan⁻¹(6 ÷ 4) ≈ 56.3°, edge tan⁻¹(6 ÷ 5.66) ≈ 46.7°.

Show me the steps
  1. Base 8, height 6: the face's horizontal leg is half the side, 4, so the angle is tan⁻¹(6 ÷ 4) ≈ 56.3°. The edge's horizontal leg is half the base diagonal, 5.66, so its angle is tan⁻¹(6 ÷ 5.66) ≈ 46.7°.
  2. Make the base 12 with height 6: the face leg is 6 and the angle is 45.0°; the edge leg is 8.49 and the angle is 35.3°. A wider base makes both flatter.
  3. Make the base 6 with height 12: the face angle is tan⁻¹(12 ÷ 3) ≈ 76.0° and the edge angle tan⁻¹(12 ÷ 4.24) ≈ 70.5°. A tall thin pyramid has both angles steep.
  4. At every setting the face angle is bigger than the edge angle, because the face's horizontal leg, a ÷ 2, is shorter than the edge's, a × √2 ÷ 2, and the height is shared.

The sloping edge is not the line that gives the angle of the face. The edge from the apex to a corner makes a smaller angle with the base than the slant of a face does. If you use the edge to find how steep a face is, the answer comes out too small.

Worked examples

1. A crate is 8 m long, 6 m wide and 5 m high. Find the longest rod that fits and the angle it makes with the floor, to 1 decimal place.

  1. Floor diagonal: √(8² + 6²) = √100 = 10.
  2. Rod: √(10² + 5²) = √125 = 11.2 m.
  3. Angle with the floor: tan−1(5 ÷ 10) = 26.6°.

2. A square-based pyramid has base side 8 cm and height 9 cm, apex above the centre. Find the angle of a face with the base and of an edge with the base, to 1 decimal place.

  1. Face: the horizontal leg is half the side, 4. tan−1(9 ÷ 4) = 66.0°.
  2. Edge: the horizontal leg is half the base diagonal, √(4² + 4²) = 5.66. tan−1(9 ÷ 5.66) = 57.8°.

3. Prove that in every square-based pyramid with its apex above the centre, the faces are steeper than the edges.

  1. Let the base side be a and the height h. For a face, tan θ = h ÷ (a/2). For an edge, tan φ = h ÷ (a√2/2).
  2. The denominators satisfy a/2 < a√2/2, because 1 < √2 and a > 0.
  3. The same h divided by a smaller number is larger, so tan θ > tan φ. Both angles are acute and tangent increases on acute angles, so θ > φ.

Your go. The first step is given. A cuboid is 10 cm long, 4 cm wide and 3 cm high. Find the angle the longest diagonal makes with the base, to 1 decimal place.

  1. The diagonal of the base is √(10² + 4²) = √116.
  2. The height is opposite the angle and the base diagonal is adjacent. Which ratio links these two?
Check the answer

tan−1(3 ÷ √116) = tan−1(0.2785) = 15.6°. Using 10 as the adjacent side, tan−1(3 ÷ 10) = 16.7°, would measure the angle with an edge of the floor, not with the base diagonal.

With a GDC (optional)

Keep a floor diagonal as an exact square root until the last step. Type the whole expression, such as the square root of (13² + 6²), in one go, or store the floor diagonal. Copying a rounded 13.04 forward into a second calculation can shift the last digit.

For the angle, use tan−1 of (height ÷ the horizontal distance) with the ratio in brackets, and check the display is in degrees (DEG). Everything here can be done with a scientific calculator.

Check before you go on

1. A cuboid is 9 cm long, 12 cm wide and 8 cm high. Which expression gives the angle between a space diagonal and the base?

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 rectangular store room in Lagos is 7 m long, 4 m wide and 3 m high. What is the length of the longest straight rod that can lie inside it, to 1 decimal place?

3. A cuboid has a base 10 cm by 6 cm and a height of 4 cm. Find the angle between a space diagonal and the base, to 1 decimal place.

Level 2 · Building

4. A square-based pyramid has a base of side 10 cm and a height of 7 cm, with its apex directly above the centre of the base. Find the angle between a sloping face and the base, to 1 decimal place.

5. A pyramid has a rectangular base 12 cm by 8 cm. Its apex is 9 cm vertically above the centre of the base. Find the length of a sloping edge, to 1 decimal place.

6. A cuboid ABCDEFGH has a base ABCD with AB = 9 cm and BC = 5 cm. The vertical edges AE, BF, CG and DH are each 4 cm. Find the angle between the plane ABGH and the base ABCD, to 1 decimal place.

Level 3 · Stretch

7. A cuboid ABCDEFGH has AB = 9 cm, BC = 5 cm and height 4 cm, with E above A, F above B, G above C and H above D. Find the angle between the line AG and the plane ABFE, to 1 decimal place.

8. VABCD is a pyramid with a square base of side 12 cm. V is directly above the centre of the base and each sloping edge is 15 cm. Find the angle between the face VAB and the base, to 1 decimal place.

Something to think about

A pyramid has a rectangular base 12 by 4 and a height of 3, with its apex above the centre. Put these in order of steepness: the faces on the long sides, the faces on the short sides, and the sloping edges. Is an edge still the least steep?

One way to think about it

Faces on the long sides: the horizontal leg is half the width, 2, so tan−1(3 ÷ 2) ≈ 56.3°. Faces on the short sides: leg 6, tan−1(3 ÷ 6) ≈ 26.6°. Edges: leg half the diagonal, √40 ≈ 6.32, tan−1(3 ÷ 6.32) ≈ 25.4°. The edge is still the least steep, because half the diagonal is longer than half of either side, but only just.

Exit ticket

9. VABCD is a pyramid with a rectangular base ABCD. V is directly above the centre O of the base, M is the midpoint of AB and N is the midpoint of BC. Which angle is the angle between the face VBC and the base ABCD?