Mathematics questions, chapter 7

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Plans and Elevations

42 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Use the measurements marked on each diagram. For drawing questions, write what the drawing shows (shape, lengths, type of line); sketch it on paper and compare with the marking scheme. Show your working for Section B.

Section A: objective questions

11 marks

Choose the correct answer. Each question carries 1 mark.

Question 1

What is a normal to a plane?

A straight line parallel to the plane
A straight line perpendicular to the plane
A straight line that lies on the plane
A straight line inclined at 45° to the plane

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A normal to a plane is a straight line that is perpendicular to the plane, so it makes a right angle with any line on the plane.

For the teacher: Remembering, low difficulty, standard 7.1.1

Textbook p. 170

Question 2

In Diagram 1, PR and QS are normals to the horizontal plane ABCD. Which line is the orthogonal projection of the line PQ on the plane ABCD?

Line PQ above a horizontal plane ABCD; dashed normals PR and QS meet the plane at R and S — Diagram 1

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The orthogonal projection of a line is found by joining the feet of the normals from its two ends. The feet are R and S, so the projection is RS.

For the teacher: Understanding, low difficulty, standard 7.1.1

Textbook p. 171

Question 3

Diagram 2 shows a cuboid ABCDEFGH on a horizontal plane. Which line is a normal to the plane BCGF?

Cuboid ABCDEFGH with base ABCD and the top face EFGH directly above it — Diagram 2

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AB meets the plane BCGF at B at a right angle. BC, BF and FG all lie on the plane BCGF, so they are not normals.

For the teacher: Understanding, low difficulty, standard 7.1.1

Textbook p. 170

Question 4

A rectangular card 8 cm by 5 cm is tilted so that its 8 cm sides stay horizontal and its 5 cm sides are inclined to the horizontal plane. Which statement describes the orthogonal projection of the card on the horizontal plane, seen from above?

A rectangle 8 cm by 5 cm
A parallelogram with sides 8 cm and 5 cm
A rectangle 8 cm long, with the other side shorter than 5 cm
A rectangle 5 cm long, with the other side shorter than 8 cm

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Sides parallel to the plane keep their length, so the 8 cm sides stay 8 cm. The inclined 5 cm sides become shorter. The right angles stay right angles, so the projection is a rectangle.

For the teacher: Understanding, medium difficulty, standard 7.1.2

Textbook p. 180

Question 5

In an elevation, a side of the object that is hidden from the viewing direction is drawn as ______.

a dashed line
a thick solid line
a thin construction line
a curved line

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Hidden sides are dashed. Visible sides are thick solid lines and thin solid lines are only construction lines.

For the teacher: Remembering, low difficulty, standard 7.2.1

Textbook p. 183

Question 6

Diagram 3 shows a block with a channel 2 cm wide and 1 cm deep along its whole top. Which drawing is the front elevation as viewed from X?

A block 6 cm by 4 cm by 3 cm with a channel 2 cm wide and 1 cm deep in the middle of the top, viewed from X in front — Diagram 3

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From X you see the end of the block: 6 cm wide and 3 cm high, with the channel as a notch in the top. A is the plan; B is the side elevation as viewed from Y; C puts the channel at the corner.

For the teacher: Applying, low difficulty, standard 7.2.1

Textbook p. 183

Question 7

In method 1 for drawing on four quadrants, the side elevation is viewed from the right. Where is it drawn?

Above the front elevation
To the left of the front elevation
To the right of the front elevation
Below the plan

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In method 1 the side elevation is in the second quadrant, on the left of the front elevation. In method 2 (viewed from the left) it is on the right.

For the teacher: Understanding, low difficulty, standard 7.2.1

Textbook p. 183

Question 8

Diagram 4 shows the plan, front elevation and side elevation of an object. What is the object?

Plan: rectangle 8 cm by 5 cm with a line down the middle. Front elevation: a pentagon, 3 cm walls and a 3 cm high triangle on top. Side elevation: rectangle 5 cm by 6 cm with a line at 3 cm. — Diagram 4

A cuboid
A triangular prism placed on a cuboid
A pyramid placed on a cuboid
A cone placed on a cylinder

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The front elevation is a rectangle with a triangle on top. The plan has one straight line (the ridge) and the side elevation is a rectangle, so the roof is a triangular prism. A pyramid would show diagonals in the plan and a triangle in the side elevation. A cone and cylinder would show a circle.

For the teacher: Analysing, high difficulty, standard 7.2.2

Textbook p. 187

Question 9

Diagram 5 shows a block with a channel along its whole top. Calculate the volume of the block.

Block 6 cm by 4 cm by 3 cm with a channel 2 cm wide and 1 cm deep along the top — Diagram 5

72 cm³
80 cm³
64 cm³
8 cm³

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Block 6 × 4 × 3 = 72 cm³. The channel is 2 × 1 × 4 = 8 cm³. Volume = 72 − 8 = 64 cm³. 72 forgets the channel; 80 adds it; 8 is the channel only.

For the teacher: Applying, low difficulty, standard 7.2.3

Textbook p. 190

Question 10

Diagram 6 shows a cone placed on a cylinder, both with diameter 6 cm. Which is the plan of the object?

A cone of height 4 cm on top of a cylinder of height 5 cm and diameter 6 cm — Diagram 6

A circle of diameter 6 cm with a dot at its centre
A circle of diameter 6 cm with a triangle inside
A rectangle 6 cm by 5 cm
A triangle on top of a rectangle

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Seen from above, both solids are circles of the same diameter, so only one circle shows. The apex of the cone is directly above the centre, so it is a dot. A rectangle or a triangle on a rectangle is a front or side elevation.

For the teacher: Applying, low difficulty, standard 7.1.1

Textbook p. 176

Question 11

Which statement about orthogonal projections is false?

The length of a side may change with the viewing direction.
A side parallel to the plane keeps its actual length.
The size of an angle may change with the viewing direction.
A side perpendicular to the plane is projected with its actual length.

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A side perpendicular to the plane is projected as a single point, not with its actual length. The other three statements are true.

For the teacher: Analysing, low difficulty, standard 7.1.2

Textbook p. 180

Section B: structured questions

31 marks

Answer all questions. Show your working.

Question 12

Diagram 7 shows a cuboid ABCDEFGH with a horizontal base ABCD.

Cuboid ABCDEFGH with base ABCD and top face EFGH — Diagram 7

[2 marks] (a)Name two normals to the plane ABCD.

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One normal from EA, FB, GC, HD (foot on the plane written last; AE, BF, CG, DH also accepted) [1]1
A different normal from the same set [1]1

The four upright edges meet the base at right angles. The foot of the normal on the plane is written last, so it may also be written EA, FB, GC and HD.

Tick each point you got.

[1 mark] (b)State the orthogonal projection of the line FD on the plane ABCD.

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BD [1]1

The normal from F meets the base at B. D is already on the plane. The projection joins B and D.

Tick each point you got.

For the teacher: Understanding, low difficulty, standard 7.1.1

Textbook p. 171

Question 13

Diagram 8 and Diagram 9 each show a triangular object and the lines that project it on a vertical plane.

A triangle with horizontal dashed lines to a vertical plane — Diagram 8

A triangle with slanted dashed lines to a vertical plane — Diagram 8

[1 mark] (a)Is the projection in Diagram 9 an orthogonal projection?

AYesBNo

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The projecting lines are slanted.

[1 mark] (b)Give a reason for your answer to (a).

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The projecting lines are not perpendicular (not normal) to the plane [1]1

Tick each point you got.

For the teacher: Understanding, low difficulty, standard 7.1.1

Textbook p. 172

Question 14

Diagram 10 shows a cylinder of diameter 6 cm and height 5 cm standing on a horizontal plane.

[1 mark] (a)State the shape and the size of the orthogonal projection on the horizontal plane, viewed from above.

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A circle of diameter 6 cm [1]1

Tick each point you got.

[2 marks] (b)State the shape and the size of the orthogonal projection on a vertical plane, viewed from the side.

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A rectangle [1]1
6 cm wide and 5 cm high [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 7.1.1

Textbook p. 176

Question 15

Diagram 11 shows a right prism PQRSTU. The base PQRS is a rectangle on a horizontal plane and the cross section PQT is a right-angled triangle with PQ = 3 cm and PT = 4 cm. PS = 8 cm.

A wedge-shaped right prism PQRSTU, base 3 cm by 8 cm, vertical edge PT 4 cm, with viewing directions X, Y and Z — Diagram 11

[2 marks] (a)Describe the plan of the prism, viewed from Z.

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A rectangle [1]1
3 cm by 8 cm [1]1

Tick each point you got.

[2 marks] (b)Describe the front elevation of the prism, viewed from X.

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A right-angled triangle [1]1
With perpendicular sides 3 cm and 4 cm (hypotenuse 5 cm) [1]1

Tick each point you got.

[2 marks] (c)The sloping edge TQ is 5 cm long. State the length of TQ in the plan and give a reason why it is not 5 cm.

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3 cm (T is drawn on P, so TQ becomes PQ) [1]1
TQ is inclined to the horizontal plane, so its projection is shorter [1]1

Tick each point you got.

For the teacher: Analysing, medium difficulty, standard 7.1.2

Textbook p. 181

Question 16

Diagram 12 shows a cuboid 6 cm by 4 cm by 3 cm with a small block, 2 cm by 2 cm and 1.5 cm high, placed in the centre of its top face.

A cuboid with a small block on the centre of its top face, with viewing directions X, Y and Z — Diagram 12

[2 marks] (a)Describe the plan viewed from Z.

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A rectangle 6 cm by 4 cm [1]1
With a 2 cm by 2 cm square in the centre [1]1

Tick each point you got.

[2 marks] (b)Describe the front elevation viewed from X.

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A rectangle 6 cm by 3 cm [1]1
With a rectangle 2 cm wide and 1.5 cm high, centred, on top [1]1

Tick each point you got.

[2 marks] (c)Calculate the volume of the object.

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6 × 4 × 3 = 72 and 2 × 2 × 1.5 = 6 [1]1
72 + 6 = 78 cm³ [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 7.2.1

Textbook p. 182

Question 17

Diagram 13 shows the plan, the front elevation and the side elevation of a solid. The side elevation is viewed from the right.

Plan: rectangle 6 cm by 3 cm with a line at 3 cm. Front elevation: an L shape 6 cm wide, 4 cm high on the left half and 2 cm high on the right half. Side elevation: rectangle 3 cm by 4 cm with a line at 2 cm. — Diagram 13

[1 mark] (a)State the height of the tallest part of the solid.

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4 cm [1]1

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[1 mark] (b)Describe the solid.

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A prism with an L-shaped cross section (or a 3 cm by 3 cm by 4 cm block joined to a 3 cm by 3 cm by 2 cm block) [1]1

Tick each point you got.

[2 marks] (c)Calculate the volume of the solid.

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Area of front elevation = 6 × 4 − 3 × 2 = 18 cm² (or 3 × 4 + 3 × 2 = 18) [1]1
Volume = 18 × 3 = 54 cm³ [1]1

Tick each point you got.

For the teacher: Analysing, high difficulty, standard 7.2.2

Textbook p. 187

Question 18

Diagram 14 shows the plan, front elevation and side elevation of a solid. The solid was a cuboid 8 cm × 6 cm × 5 cm. A right prism with a triangular cross section was removed from one top edge.

Plan: rectangle 8 cm by 6 cm with a line at 5 cm. Front elevation: a rectangle with the top right corner cut off by a sloping line; top 5 cm, horizontal part cut 3 cm, vertical part cut 4 cm, remaining right edge 1 cm. Side elevation: rectangle 6 cm by 5 cm with a line at 1 cm. — Diagram 14

[2 marks] (a)Calculate the length, in cm, of the sloping edge in the front elevation.

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3² + 4² = 25 [1]1
5 cm [1]1

Tick each point you got.

[2 marks] (b)Calculate the volume, in cm³, of the prism that was removed.

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½ × 3 × 4 × 6 [1]1
= 36 cm³ [1]1

Tick each point you got.

[2 marks] (c)Calculate the volume, in cm³, of the solid shown.

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8 × 6 × 5 = 240 [1]1
240 − 36 = 204 cm³ [1]1

Tick each point you got.

[1 mark] (d)Find the ratio of the volume removed to the volume of the solid shown, in its simplest form.

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36 : 204 = 3 : 17 [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 7.2.3

Textbook p. 190

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