AQA Level 2 Further Maths · Level 2 Certificate · Chapter 7: Geometry II · Lesson 4

Problems in three dimensions

About 90 minutesCalculator allowed63 marks

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Date

Do Now

Answer these before the lesson starts.

A1
Needed today
A cuboid measures $8$ cm by $9$ cm by $12$ cm. Work out the length of a diagonal from one corner to the opposite corner.
[2 marks]
Answer $\dots\dots$ cm
A2
Needed today
In a right-angled triangle, the side opposite angle $θ$ is $7.2$ cm and the other shorter side is $5$ cm. Work out $θ$ .
[2 marks]
Answer $θ = \dots \dots^{\circ}$
A3
Last lesson
A triangle has sides of $5$ cm, $7$ cm and $9$ cm. Work out the size of its largest angle.
[2 marks]
Answer $\dots \dots^{\circ}$
A4
From chapter 7
A triangle has sides of $11$ cm and $14$ cm with an angle of $3 5^{\circ}$ between them. Work out its area.
[2 marks]
Answer $\dots\dots$ cm²

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it. Give lengths and angles to 3 significant figures. In each cuboid, $E$ , $F$ and $G$ are above $A$ , $B$ and $C$ .

Example 1

The diagram shows a cuboid. Work out the size of angle $E B G$ in the triangle $B E G$ .

Not drawn accurately

Hint Each side is a diagonal of a face: find all three by Pythagoras. Then use the cosine rule.

B1
The diagram shows a cuboid. Work out the size of angle $B E G$ in the triangle $B E G$ .
[4 marks]
Not drawn accurately
Answer $\dots \dots^{\circ}$

Example 2

Pyramid $V A B C D$ has a horizontal rectangular base. $V$ is above its centre $X$ and $V X = 8$ cm. $M$ is the midpoint of $B C$ . Work out the angle between the base and (a) the face $V B C$ (b) the edge $V B$ .

Not drawn accurately

Hint The planes meet along

B C

: use

V M

and

X M

, both at right angles to

B C

X M

is half of

A B

. For the edge, use

X B

B2
Pyramid $V A B C D$ has a horizontal rectangular base. $V$ is above its centre $X$ and $V X = 11$ cm. Work out the angle between the base and (a) the face $V A B$ (b) the face $V B C$ .
[4 marks]
Not drawn accurately
Answer(a) $\dots \dots^{\circ}$ (b) $\dots \dots^{\circ}$

Example 3

A level road $A B$ is $120$ m long. $X T$ is a vertical mast with its foot $X$ level with the road. Angle $X A B = 3 4^{\circ}$ , angle $X B A = 5 8^{\circ}$ and the angle of elevation of $T$ from $A$ is $2 1^{\circ}$ . Work out (a) the height $X T$ (b) the angle of elevation of $T$ from $B$ .

Not drawn accurately

Hint On the ground, use the sine rule in triangle

A B X

. Each triangle with

X T

in it is right-angled at

X

B3
A level road $P Q$ is $250$ m long. $F T$ is a vertical mast with its foot $F$ level with the road. Angle $F P Q = 4 1^{\circ}$ , angle $F QP = 6 3^{\circ}$ and the angle of elevation of $T$ from $Q$ is $1 8^{\circ}$ . Work out (a) the height $F T$ (b) the angle of elevation of $T$ from $P$ .
[5 marks]
Not drawn accurately
Answer(a) $\dots\dots$ m (b) $\dots \dots^{\circ}$

Practice

Level 2

C1
The diagram shows a wedge with a horizontal rectangular base $A B C D$ and a vertical rectangular face $C D E F$ . Work out (a) angle $F B C$ (b) the angle between $A F$ and the base $A B C D$ .
[4 marks]
Not drawn accurately
Answer(a) $\dots \dots^{\circ}$ (b) $\dots \dots^{\circ}$
C2
A basket is the shape of part of a square-based pyramid. Its top $A B C D$ is a square of side $26$ cm and its base $P QR S$ is a square of side $16$ cm. The top and the base are horizontal, with their centres on one vertical line, $30$ cm apart. Work out (a) the length $A P$ (b) the angle between $A P$ and the base $P QR S$ .
[4 marks]
Not drawn accurately
Answer(a) $\dots\dots$ cm (b) $\dots \dots^{\circ}$

C3
$A B C D$ is a tetrahedron. $A C = 6$ cm, $A D = 9$ cm and angle $C A D = 11 8^{\circ}$ . In the face $B C D$ , angle $B C D = 7 0^{\circ}$ and angle $B D C = 4 8^{\circ}$ . Work out the length $B C$ .
[4 marks]
Not drawn accurately
Answer $B C = \dots\dots$ cm

Exam-style questions

AQA exam style

D1
Two lines in three dimensions do not meet and are not parallel. Which word describes them? Circle your answer.
$parallel skew perpendicular intersecting$
[1 mark]

D2
Ella says, "The face $V B C$ contains the edge $V B$ , so the angle between the face $V B C$ and the base is the same as the angle between the edge $V B$ and the base."
$V A B C D$ is a pyramid with a horizontal square base $A B C D$ of side $10$ cm. $V$ is directly above $X$ , the centre of the base. $V A = V B = V C = V D = 13$ cm. $N$ is the midpoint of $B C$ .
Not drawn accurately
(a)
Show that the height $V X$ is $119$ cm.
[2 marks]
(b)
Work out the angle between the edge $V A$ and the base.
[2 marks]
Answer $\dots \dots^{\circ}$
(c)
Show that Ella is wrong.
[3 marks]
Total for D2: 7 marks
D3
$V A B C D$ is a pyramid with a horizontal rectangular base $A B C D$ . $V$ is directly above the centre of the base. $A B = 14$ cm, $B C = 10$ cm and $V A = V B = V C = V D = 15$ cm. $M$ is the midpoint of $A B$ .
Not drawn accurately
(a)
Work out the size of angle $M V C$ . You must show your working.
[5 marks]
Answer $\dots \dots^{\circ}$
(b)
Work out the area of triangle $M V C$ .
[2 marks]
Answer $\dots\dots$ cm²
Total for D3: 7 marks
D4
$A$ , $B$ and $C$ are points on horizontal ground. $B$ is $180$ m from $A$ on a bearing of $07 0^{\circ}$ . The bearing of $C$ from $A$ is $02 5^{\circ}$ and the bearing of $C$ from $B$ is $33 0^{\circ}$ . $C$ is the foot of a vertical tower $C T$ . The angle of elevation of $T$ from $A$ is $1 2^{\circ}$ .
Not drawn accurately
(a)
Show that angle $A C B = 5 5^{\circ}$ and work out the distance $A C$ .
[3 marks]
Answer $A C = \dots\dots$ m
(b)
Work out the height of the tower.
[2 marks]
Answer $\dots\dots$ m
(c)
Work out the angle of elevation of $T$ from $B$ .
[2 marks]
Answer $\dots \dots^{\circ}$
Total for D4: 7 marks
D5
$X$ and $Y$ are $40$ m apart on level ground, in line with the foot $F$ of a vertical mast $F T$ . From $X$ , the angle of elevation of $T$ is $1 9^{\circ}$ . From $Y$ , it is $3 4^{\circ}$ . Work out the height of the mast.
[4 marks]
Not drawn accurately
Answer $\dots\dots$ m

Extension

Stretch

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

E1
$A B C D$ is a regular tetrahedron: each of its six edges is $12$ cm. Work out the angle between two of its faces.
[4 marks]
Not drawn accurately
Hint 1 · What to try
The faces $A B C$ and $D B C$ meet along $B C$ . Let $M$ be the midpoint of $B C$ : $A M$ and $D M$ are both at right angles to $B C$ , so the angle you want is $A M D$ .
Hint 2 · The first line
$A M$ and $D M$ are heights of equilateral triangles of side $12$ : $A M = D M = 1 2^{2} - 6^{2} = 108$ . And $A D = 12$ .
Hint 3 · The full method
$cos A M D = \frac{108 + 108 - 144}{2 \times 108} = \frac{1}{3}$ , so the angle is $70. 5^{\circ}$ (whatever the length of the edges).
Answer $\dots \dots^{\circ}$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find lengths in a solid using Pythagoras in three dimensions.
I can find the angle between a line and a plane.
I can find the angle between two planes.
I can use the sine and cosine rules in three-dimensional problems.

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