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Cambridge IGCSE0580

Pythagoras’ theorem and trigonometry in 3D

Mathematics 0580 Chapter Notes

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Pythagoras’ theorem and trigonometry in 3D
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In this chapter

  1. 1Find the right-angled triangle
  2. 2The diagonal of a cuboid
  3. 3Pyramids and the half-diagonal
  4. 4The angle between a line and a plane
  5. 5The angle of a diagonal, in one step

1. Find the right-angled triangle

Every three-dimensional problem is solved in a right-angled triangle. Find it, then draw it flat and label its sides.

The triangle may lie in a face, in the base, or inside the solid. Work in stages: first a length in the base, then a length through the solid.

A vertical edge makes a right angle with every line in the base.

a² + b² = c²

Worked example 13 marks

Do not use a calculator. The base of a cuboid is a rectangle 15 cm by 8 cm. Find the length of the diagonal of the base.

  1. 1

    diagonal² = 15² + 8² [1]

  2. 2

    = 225 + 64 = 289 [1]

  3. 3

    diagonal = √(289) = 17 cm [1]

2. The diagonal of a cuboid

The longest diagonal joins opposite corners of a cuboid. It is the hypotenuse of a triangle whose other sides are the diagonal of the base and the height.

Square all three edges, add the squares, then take the square root.

d² = l² + w² + h²

Examiner insight

“This was a challenge to some candidates as a diagonal of a face was occasionally given as a final answer.” “Some less able candidates added the lengths of two or three sides of the cuboid.”

Worked example 13 marks

ABCDEFGH is a cuboid. AB = 18 cm, BC = 14 cm and AE = 16 cm. Calculate the length of the diagonal AG.

  1. 1

    AG² = 18² + 14² + 16² [1]

  2. 2

    = 324 + 196 + 256 = 776 [1]

  3. 3

    AG = √(776) = 27.9 cm [1]

3. Pyramids and the half-diagonal

When the apex of a square-based pyramid is vertically above the centre of the base, the sloping edge, the height and the line from the centre to a corner form a right-angled triangle.

The line from the centre to a corner is half of the length of the diagonal of the base. Find the base diagonal first, then halve it.

Examiner insight

“Candidates are advised that the requirement on the front of the question paper is to give non-exact answers correct to three significant figures in order to gain the accuracy marks.”

Worked example 14 marks

ABCDE is a square-based pyramid. EC and BD intersect at F. AF is perpendicular to the base BCDE. BC = 16.5 cm and AF = 28 cm. Calculate AB.

  1. 1

    BD² = 16.5² + 16.5² = 544.5 [1]

  2. 2

    BF = 1/2BD, so BF² = 544.5 ÷ 4 = 136.125 [1]

  3. 3

    AB² = 28² + 136.125 = 920.125 [1]

  4. 4

    AB = √(920.125) = 30.3 cm [1]

4. The angle between a line and a plane

To find the angle between a line and a plane, drop a perpendicular from the top of the line to the plane. Join the foot of the perpendicular to the point where the line meets the plane. The angle is between the line and this joining line.

The angle is not an angle in a face. Draw the right-angled triangle, then use sine, cosine or tangent.

sin x = opp/hyp

tan x = opp/adj

Examiner insight

“Candidates who added to the diagram often met with more success.”

Worked example 12 marks

ABCDE is a square-based pyramid. EC and BD intersect at F. AF is perpendicular to the base BCDE. BC = 16.5 cm and AF = 28 cm. Calculate the angle between AB and the base BCDE.

  1. 1

    AF is perpendicular to the base, so the angle is ABF. BF is half the diagonal of the base: BF = √(136.125) = 11.667... cm

  2. 2

    tan ABF = 28/11.667... [1]

  3. 3

    ABF = 67.4° [1]

5. The angle of a diagonal, in one step

For a cuboid, the height is the side opposite the angle and the diagonal is the hypotenuse. So sine needs no base diagonal.

Keep the unrounded value until the last line. Give an angle correct to 1 decimal place.

Examiner insight

“A few candidates demonstrated a correct method but rounded during their working and gave a final answer out of the required range of accuracy.”

Worked example 13 marks

A cuboid measures 8 cm by 5 cm by 3 cm. Calculate the angle between its longest diagonal and the base.

  1. 1

    d² = 8² + 5² + 3² = 98 [1]

  2. 2

    sin x = 3/(√98) [1]

  3. 3

    x = 17.6° [1]

Recap

  • Find a right-angled triangle, then draw it flat.
  • Cuboid diagonal: d² = l² + w² + h².
  • Square-based pyramid with its apex above the centre: the distance from the centre to a corner is half the base diagonal.
  • Angle between a line and a plane: drop a perpendicular to the plane and join its foot to the point where the line meets the plane.
  • Not an angle in a face. Add to the diagram.
  • Do not round until the last line. Three significant figures for lengths, one decimal place for angles.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. A cuboid measures 2 cm by 3 cm by 6 cm. Find the length of its longest diagonal.2 marks
  2. Do not use a calculator. The base of a cuboid is 8 cm by 6 cm and its height is 24 cm. Find the length of the diagonal from one corner of the base to the opposite corner of the top.4 marks
  3. A room is 5.2 m long, 4.1 m wide and 2.8 m high. A straight cable runs from one corner of the floor to the opposite corner of the ceiling. Calculate the length of the cable.3 marks
  4. Do not use a calculator. A pyramid has a square base of side 8 cm. Its apex is vertically above the centre of the base, at a height of 2 cm. Find the length of a sloping edge.4 marks
  5. A cuboid is 12 cm long, 9 cm wide and 8 cm high. Calculate the angle between its longest diagonal and the base.4 marks
  6. Do not use a calculator. Each sloping edge of a square-based pyramid is 12 cm long and makes an angle of 30° with the base. Find the height of the pyramid.2 marks
  7. A vertical mast, 15 m high, stands at corner P of a horizontal rectangular field PQRS. PQ = 80 m and QR = 60 m. Calculate the angle between the line from R to the top of the mast and the ground.4 marks
  8. Do not use a calculator. A cube has edges of 4 cm. Find the length of its longest diagonal. Give your answer as a surd in its simplest form.3 marks
  9. Do not use a calculator. A pyramid has a rectangular base 12 cm by 16 cm. Its apex is vertically above the centre of the base. Each sloping edge is 26 cm long. (a) Find the height of the pyramid. (b) The angle between a sloping edge and the base is x°. Find the value of sin x°, as a fraction in its simplest form. (c) M is the midpoint of a 12 cm side of the base. The angle between the base and the line from the apex to M is y°. Find the value of tan y°.8 marks
  10. A room is a cuboid. Its floor is a rectangle 12 m by 9 m and the room is 5 m high. A lamp hangs 1.4 m vertically below the centre of the ceiling. A straight wire runs from a corner A of the floor to the lamp. (a) Calculate the length of the wire. (b) Calculate the angle between the wire and the floor. (c) The longest diagonal of the room also starts at A. Calculate the angle between the wire and this diagonal.9 marks

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