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GCSE · Geometry and Measures

4.13D Volume of a Prism: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculator allowed · about 85 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Formulae You May Need

Revision Library formulae reminder, styled on Edexcel 1MA1 formulae-sheet conventions; wording is original.

Volume of a prism = area of cross section x length

Area of a trapezium = (1/2) x (a + b) x h

Note: on the Edexcel formulae sheet, both of these formulae are given to you (foundation and higher tiers). You still need to recall the area of a rectangle (A = l x w), the area of a triangle (A = (1/2) x b x h) and the area of a parallelogram (A = b x h) yourself - these are not given.

1 litre = 1000 cm^3

1 m^3 = 1,000,000 cm^3 (since 1 m = 100 cm, and 100^3 = 1,000,000)

A cuboid has length 9 cm, width 6 cm and height 4 cm. Work out the volume of the cuboid.

Diagram NOT accurately drawn

(Total for Question 1 is 2 marks)

A cube has edge length 9 cm. Work out the volume of the cube.

Diagram NOT accurately drawn

(Total for Question 2 is 2 marks)

The cross-section of a prism is an irregular hexagon with area 18.5 cm^2. The prism has length 12 cm. Work out the volume of the prism.

(Total for Question 3 is 2 marks)

A cuboid has length 7.5 cm, width 4 cm and height 6 cm. Work out the volume of the cuboid.

Diagram NOT accurately drawn

(Total for Question 4 is 2 marks)

A cube has volume 64 cm^3. Work out the edge length of the cube.

(Total for Question 5 is 2 marks)

A triangular prism has a cross-section that is a right-angled triangle with base 8 cm and perpendicular height 5 cm. The prism has length 11 cm. Work out the volume of the prism.

Diagram NOT accurately drawn

(Total for Question 6 is 3 marks)

A container holds 4500 cm³ of water. Work out this volume in litres.

(Total for Question 7 is 2 marks)

A prism has a cross-section in the shape of a trapezium. The trapezium has parallel sides of length 6 cm and 10 cm, and the perpendicular distance between the parallel sides is 4 cm. The prism has length 12 cm. Calculate the volume of the prism.

Diagram NOT accurately drawn

(Total for Question 8 is 3 marks)

A storage tank has a volume of 2,500,000 cm^3. Work out this volume in m^3.

(Total for Question 9 is 2 marks)

A cuboid has volume 360 cm^3. The cuboid has length 10 cm and width 6 cm. Work out the height of the cuboid.

Diagram NOT accurately drawn

(Total for Question 10 is 3 marks)

A triangular prism has a cross-section that is a right-angled triangle with base 7 cm and perpendicular height 6 cm. The volume of the prism is 315 cm^3. Work out the length of the prism.

Diagram NOT accurately drawn

(Total for Question 11 is 3 marks)

A prism has a cross-section in the shape of a parallelogram. The parallelogram has base 9 cm and perpendicular height 5 cm. The prism has length 14 cm. Calculate the volume of the prism.

Diagram NOT accurately drawn

(Total for Question 12 is 3 marks)

A solid metal cuboid has dimensions 5 cm by 4 cm by 3 cm. The density of the metal is 8.9 g/cm^3. Work out the mass of the cuboid.

Diagram NOT accurately drawn

(Total for Question 13 is 3 marks)

The diagram shows the cross-section of a prism. The cross-section is made from a rectangle measuring 12 cm by 9 cm, with a rectangular notch measuring 5 cm by 4 cm removed from one corner. The prism has length 7 cm. Work out the volume of the prism.

Diagram NOT accurately drawn

(Total for Question 14 is 4 marks)

A builder is making a concrete garden edging trough shaped like a triangular prism. The triangular cross-section is a right-angled triangle with base 30 cm and perpendicular height 20 cm. The trough is 200 cm long.

Diagram NOT accurately drawn

(a)Work out the volume of concrete needed to make the trough. Give your answer in cm^3.(3)

(b)Concrete is sold in bags that each contain 25000 cm³ of concrete. Work out the minimum number of complete bags the builder must buy to have enough concrete for the trough.(2)

(Total for Question 15 is 5 marks)

A fish tank is a cuboid with length 60 cm, width 30 cm and height 40 cm.

Diagram NOT accurately drawn

(a)Work out the volume of the tank. Give your answer in litres. (1 litre = 1000 cm³)(2)

(b)Layla fills the tank using a hose that delivers water at a constant rate of 0.8 litres per second. Work out the time taken to fill the tank completely. Give your answer in minutes and seconds.(2)

(Total for Question 16 is 4 marks)

A factory makes solid rubber doorstops in the shape of a triangular prism. Each doorstop has a cross-section that is a right-angled triangle with base 6 cm and perpendicular height 4 cm, and the prism length is 11 cm.

Diagram NOT accurately drawn

(a)Show that the volume of one doorstop is 132 cm^3.(2)

(b)The factory has a block of rubber with volume 3.168 m^3. Work out the maximum number of complete doorstops that can be made from this block of rubber, assuming there is no wastage.(3)

(Total for Question 17 is 5 marks)

Two storage boxes are in the shape of cuboids. Box A has dimensions 40 cm by 35 cm by 50 cm. Box B has dimensions 30 cm by 45 cm by 55 cm. Work out which box has the greater volume, and work out by how much.

Diagram NOT accurately drawn

(Total for Question 18 is 5 marks)

A cuboid has length (x + 2) cm, width 4 cm and height 5 cm. The volume of the cuboid is 180 cm^3. Form an equation in terms of x, and solve it to find the value of x.

Diagram NOT accurately drawn

(Total for Question 19 is 3 marks)

A swimming pool is built in the shape of a prism. Looking at the pool from the side, the cross-section is a trapezium: the pool is 1.2 m deep at the shallow end and 2.4 m deep at the deep end, with the floor sloping uniformly between the two ends. The distance from the shallow end to the deep end is 25 m, and the pool is 10 m wide.

Diagram NOT accurately drawn

(a)Work out the cross-sectional area of the pool (the trapezium formed by the sloping floor).(2)

(b)Work out the volume of water needed to fill the pool completely. Give your answer in m^3.(2)

(c)Given that 1 m³ = 1000 litres, work out the volume of water in litres.(1)

(Total for Question 20 is 5 marks)

A cuboid has length (x + 3) cm, width x cm and height 4 cm. The volume of the cuboid is 280 cm^3.

Diagram NOT accurately drawn

(a)Show that x² + 3x - 70 = 0.(3)

(b)Hence find the value of x, giving a reason why one of the solutions to the equation should be rejected.(3)

(Total for Question 21 is 6 marks)

A solid prism has length 20 cm. Its cross-section (shown in the diagram) is a pentagon made from a rectangle with a triangle on top, like the end of a house shape. The rectangular part has width 8 cm and height 6 cm. The triangular part on top has a horizontal base of 8 cm (the same as the rectangle's width), and each of its two sloping sides has length 5 cm.

Diagram NOT accurately drawn

(a)Show that the perpendicular height of the triangular part of the cross-section is 3 cm.(3)

(b)Work out the total volume of the solid prism.(4)

(c)The solid prism is melted down and recast, with no wastage, into identical small cubes of edge length 2 cm. Work out how many complete small cubes can be made.(3)

(Total for Question 22 is 10 marks)

Mark scheme · 4.13D Volume of a Prism: Fluency and Exam Drill

Question 1

M1 correct substitution into V = l x w x h, e.g. 9 x 6 x 4 (oe)
A1 216 cm³ (cao)
Answer: 216 cm³

Question 2

M1 correct substitution into V = l³, e.g. 9³ (oe)
A1 729 cm³ (cao)
Answer: 729 cm³

Question 3

M1 correct substitution into V = area of cross section x length, e.g. 18.5 x 12 (oe)
A1 222 cm³ (cao)
Answer: 222 cm³

Question 4

M1 correct substitution into V = l x w x h, e.g. 7.5 x 4 x 6 (oe)
A1 180 cm³ (cao)
Answer: 180 cm³

Question 5

M1 correct method to find a cube root, e.g. cube root of 64 (oe)
A1 4 cm (cao)
Answer: 4 cm

Question 6

M1 area of triangle = (1/2) x 8 x 5 (oe), = 20 cm²
M1 correct substitution into V = area of cross section x length, e.g. 20 x 11 (oe, ft their area)
A1 220 cm³ (cao)
Answer: 220 cm³

Question 7

M1 correct use of 1 litre = 1000 cm³, e.g. 4500 / 1000 (oe)
A1 4.5 litres (cao)
Answer: 4.5 litres

Question 8

M1 area of trapezium = (1/2) x (6 + 10) x 4 (oe), = 32 cm²
M1 correct substitution into V = area of cross section x length, e.g. 32 x 12 (oe, ft their area)
A1 384 cm³ (cao)
Answer: 384 cm³

Question 9

M1 correct use of 1 m³ = 1,000,000 cm³, e.g. 2500000 / 1000000 (oe)
A1 2.5 m³ (cao)
Answer: 2.5 m³

Question 10

M1 forms 360 = 10 x 6 x h (oe)
M1 10 x 6 = 60 (oe), leading to h = 360 / 60
A1 6 cm (cao)
Answer: 6 cm

Question 11

M1 area of triangle = (1/2) x 7 x 6 (oe), = 21 cm²
M1 forms length = 315 / 21 (ft their area)
A1 15 cm (cao)
Answer: 15 cm

Question 12

M1 area of parallelogram = 9 x 5 (oe), = 45 cm²
M1 correct substitution into V = area of cross section x length, e.g. 45 x 14 (oe, ft their area)
A1 630 cm³ (cao)
Answer: 630 cm³

Question 13

M1 volume = 5 x 4 x 3 (oe), = 60 cm³
M1 mass = volume x density, e.g. 60 x 8.9 (oe, ft their volume)
A1 534 g (cao)
Answer: 534 g

Question 14

M1 area of large rectangle = 12 x 9 (oe), = 108 cm²
M1 area of notch = 5 x 4 (oe), = 20 cm²
M1 area of cross section = 108 - 20 = 88 cm², then x length, e.g. 88 x 7 (oe, ft their area)
A1 616 cm³ (cao)
Answer: 616 cm³

Question 15

(a) M1 area of triangle = (1/2) x 30 x 20 (oe), = 300 cm²
(a) M1 correct substitution into V = area of cross section x length, e.g. 300 x 200 (oe, ft their area)
(a) A1 60000 cm³ (cao)
(a) Answer: 60000 cm³
(b) M1 60000 / 25000 (oe, ft their volume from part (a)), = 2.4
(b) A1 3 bags (cao - must round up, since 2 bags would not contain enough concrete)
(b) Answer: 3 bags

Question 16

(a) M1 volume = 60 x 30 x 40 (oe), = 72000 cm³
(a) A1 72 litres (cao)
(a) Answer: 72 litres
(b) M1 time in seconds = 72 / 0.8 (oe, ft their litres from part (a)), = 90
(b) A1 1 minute 30 seconds (cao)
(b) Answer: 1 minute 30 seconds

Question 17

(a) M1 area of triangle = (1/2) x 6 x 4 = 12 cm²
(a) A1 12 x 11 = 132 cm³ fully shown with no errors (cso)
(a) Answer: 132 cm³ (shown)
(b) M1 converts 3.168 m³ to cm³, e.g. 3.168 x 1,000,000 (oe), = 3,168,000 cm³
(b) M1 3,168,000 / 132 (oe, ft their cm³ volume and the value from part (a))
(b) A1 24000 doorstops (cao)
(b) Answer: 24000 doorstops

Question 18

M1 volume of Box A = 40 x 35 x 50 (oe), = 70000 cm³
M1 volume of Box B = 30 x 45 x 55 (oe), = 74250 cm³
M1 valid method to compare the two volumes, e.g. subtracts the smaller from the larger
A1 4250 cm³ (cao)
B1 correct conclusion stated: Box B has the greater volume
Answer: Box B has the greater volume, by 4250 cm^3.

Question 19

M1 forms a correct equation, e.g. (x + 2) x 4 x 5 = 180, or 20(x + 2) = 180 (oe)
M1 correct rearrangement, e.g. x + 2 = 9 (oe)
A1 x = 7 (cao)
Answer: x = 7

Question 20

(a) M1 correct substitution into area of trapezium = (1/2) x (1.2 + 2.4) x 25 (oe)
(a) A1 45 m² (cao)
(a) Answer: 45 m²
(b) M1 volume = area of cross section x width, e.g. 45 x 10 (oe, ft their area from part (a))
(b) A1 450 m³ (cao)
(b) Answer: 450 m³
(c) B1 450000 litres (cao, ft their m³ from part (b))
(c) Answer: 450000 litres

Question 21

(a) M1 forms a correct volume equation, e.g. (x + 3) x (x) x 4 = 280, or 4x(x + 3) = 280 (oe)
(a) M1 correctly expands, e.g. 4x² + 12x = 280 (oe)
(a) A1 divides through by 4 and rearranges to x² + 3x - 70 = 0 fully shown with no errors (cso)
(a) Answer: x² + 3x - 70 = 0 (shown)
(b) M1 correct factorisation (x - 7)(x + 10) = 0 (oe), or correct use of the quadratic formula
(b) M1 x = 7 or x = -10 identified (both values found)
(b) A1 x = 7, with a reason given for rejecting x = -10 (e.g. a length cannot be negative)
(b) Answer: x = 7 (x = -10 is rejected, since a length cannot be negative)

Question 22

(a) M1 identifies the triangle as isosceles and uses half the base, e.g. half of 8 = 4 cm
(a) M1 correct use of Pythagoras, e.g. height = √5² - 4² (oe)
(a) A1 height = 3 cm fully shown with no errors (cso)
(a) Answer: 3 cm (shown)
(b) M1 area of rectangle = 8 x 6 (oe), = 48 cm²
(b) M1 area of triangle = (1/2) x 8 x 3 (oe, ft their height from part (a)), = 12 cm²
(b) M1 total cross-sectional area = 48 + 12 = 60 cm², then x length, e.g. 60 x 20 (oe, ft)
(b) A1 1200 cm³ (cao)
(b) Answer: 1200 cm³
(c) M1 volume of one small cube = 2³ (oe), = 8 cm³
(c) M1 1200 / 8 (oe, ft their volume from part (b) and cube volume)
(c) A1 150 cubes (cao)
(c) Answer: 150 cubes

Volume of a Prism: Fluency and Exam Drill - Worksheets, Questions and Revision

4.13D Volume of a Prism: Fluency and Exam Drill

Formulae You May Need

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20

Question 21

Question 22