Revision Library

revisionlibrary.co.uk

GCSE · Geometry and Measures

4.12 Surface Area

EDEXCEL 1MA1 · Calculator allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

A cube has edge length x.

(a)Write down a formula for the total surface area, S, of the cube in terms of x.(1)

(b)Work out the total surface area of a cube with edge length 5 cm.(2)

(Total for Question 1 is 3 marks)

The diagram shows a cuboid with length 9 cm, width 4 cm and height 3 cm.

Diagram NOT accurately drawn

(Total for Question 2 is 3 marks)

Priya makes an open-topped rectangular planter (no lid) with length 120 cm, width 40 cm and height 50 cm. Work out the total surface area of the planter, including the base but not the open top.

Diagram NOT accurately drawn

(Total for Question 3 is 3 marks)

A cube has edge length 4 cm. Which of the following is the total surface area of the cube?

A) 16 cm²
B) 64 cm²
C) 96 cm²
D) 24 cm²

(Total for Question 4 is 1 mark)

A triangular prism has a right-angled triangular cross-section with shorter sides 3 cm and 4 cm and hypotenuse 5 cm. The length of the prism is 10 cm. Work out the total surface area of the prism.

Diagram NOT accurately drawn

(Total for Question 5 is 4 marks)

A closed cylinder has radius 4 cm and height 10 cm. Work out the total surface area of the cylinder. Give your answer correct to 3 significant figures.

Diagram NOT accurately drawn

(Total for Question 6 is 3 marks)

A cylindrical bucket, open at the top, has radius 15 cm and height 25 cm. Work out the total surface area of the bucket (curved surface and base only). Give your answer correct to 3 significant figures.

Diagram NOT accurately drawn

(Total for Question 7 is 3 marks)

A solid is made from two cuboids joined to form an L-shaped prism. The L-shaped cross-section fits inside a 10 cm by 8 cm bounding rectangle, with a 6 cm by 4 cm rectangular corner removed. The prism has a length of 8 cm. Work out the total surface area of the solid.

Diagram NOT accurately drawn

(Total for Question 8 is 5 marks)

An isosceles triangular prism has a cross-section with base 6 cm and two equal sides of 5 cm. The prism has length 12 cm.

Diagram NOT accurately drawn

(a)Show that the perpendicular height of the triangular cross-section is 4 cm.(2)

(b)Work out the total surface area of the prism.(3)

(Total for Question 9 is 5 marks)

A closed cylinder has radius 6 cm and height 15 cm. Show that the total surface area of the cylinder is 792 cm², correct to 3 significant figures.

Diagram NOT accurately drawn

(Total for Question 10 is 3 marks)

A prism has a trapezium cross-section with parallel sides 4 cm and 12 cm, perpendicular height 3 cm between the parallel sides, and the two sloping sides are each 5 cm. The prism has length 10 cm. Work out the total surface area of the prism.

Diagram NOT accurately drawn

(Total for Question 11 is 4 marks)

A garden shed (excluding floor and roof) is a cuboid measuring 3 m long, 2 m wide and 2.4 m high. Dan wants to paint the four external walls only (not the roof or floor). One tin of paint covers 5 m² and costs 12.50 pounds. Work out the total cost of the paint Dan needs to buy.

(Total for Question 12 is 5 marks)

A cube has volume 343 cm^3. Work out the total surface area of the cube.

(Total for Question 13 is 3 marks)

A closed cylinder has radius 5 cm. The total surface area of the cylinder is 534 cm², correct to 3 significant figures. Work out the height of the cylinder.

Diagram NOT accurately drawn

(Total for Question 14 is 4 marks)

A small cube of edge 2 cm is cut from one corner of a larger solid cube of edge 8 cm. Work out the total surface area of the remaining solid, giving a reason for your method.

Diagram NOT accurately drawn

(Total for Question 15 is 3 marks)

Cylinder A has total surface area 150 cm^2. Cylinder B is mathematically similar to Cylinder A, with all linear dimensions 3 times as large. Work out the total surface area of Cylinder B.

(Total for Question 16 is 4 marks)

A cuboid measures 20 cm by 10 cm by 10 cm. A cylindrical hole of radius 3 cm is drilled completely through the cuboid, from the centre of one 10 cm by 10 cm face to the centre of the opposite 10 cm by 10 cm face. Work out the total surface area of the resulting solid. Give your answer correct to 3 significant figures.

Diagram NOT accurately drawn

(Total for Question 17 is 5 marks)

A sphere has radius 9 cm. Work out the total surface area of the sphere. Give your answer correct to 3 significant figures. (You may use the formula S = 4 x π x r²)

Diagram NOT accurately drawn

(Total for Question 18 is 2 marks)

A solid hemisphere has radius 6 cm. Work out the total surface area of the hemisphere (curved surface plus the flat circular face). Give your answer correct to 3 significant figures.

Diagram NOT accurately drawn

(Total for Question 19 is 3 marks)

A cone has base radius 5 cm and perpendicular height 12 cm.

Diagram NOT accurately drawn

(a)Work out the slant height, l, of the cone.(2)

(b)Work out the total surface area of the cone (curved surface plus base). Give your answer correct to 3 significant figures. (You may use the formula: curved surface area = π x r x l)(3)

(Total for Question 20 is 5 marks)

Mark scheme · 4.12 Surface Area

Question 1

(a) B1 S = 6x² oe
(a) Answer: S = 6x²
(b) M1 5² (= 25) seen, or 6 x 5²
(b) A1 150 cm² cao
(b) Answer: 150 cm²

Question 2

M1 at least two correct face areas from 36, 27, 12
M1 complete method, e.g. 2(36 + 27 + 12)
A1 150 cm² cao
Answer: 150 cm²

Question 3

M1 base area 120 x 40 (= 4800) seen
M1 complete method for the four side faces, e.g. 2(120x50) + 2(40x50) (= 16000)
A1 20800 cm² cao
Answer: 20800 cm²

Question 4

B1 C (96 cm²)
Answer: C) 96 cm²

Question 5

M1 area of triangular face = 0.5x3x4 (= 6), doubled for two triangles (= 12)
M1 three rectangular faces: 3x10, 4x10, 5x10 (= 30, 40, 50)
M1 complete method, e.g. 12 + 30 + 40 + 50
A1 132 cm² cao
Answer: 132 cm²

Question 6

M1 2 x π x 4² (= 32pi or 100.5) or 2 x π x 4 x 10 (= 80pi or 251.3) seen
M1 complete method, 2xpix4² + 2xpix4x10
A1 awrt 352 cm²
Answer: 352 cm² (3 s.f.)

Question 7

M1 π x 15² (= 225pi or 706.9) seen
M1 2 x π x 15 x 25 (= 750pi or 2356.2) seen, or complete method
A1 awrt 3060 cm²
Answer: 3060 cm² (3 s.f.)

Question 8

M1 area of L-shaped cross-section = 10x8 - 6x4 (= 80 - 24 = 56)
M1 perimeter of L-shaped cross-section = 10+4+6+4+4+8 (= 36)
M1 perimeter x length, 36 x 8 (= 288)
dM1 complete method, 2 x 56 + 288
A1 400 cm² cao
Answer: 400 cm²

Question 9

(a) M1 √5² - 3² or 5² - 3² (= 16) seen
(a) A1 4 cm cso
(a) Answer: 4 cm
(b) M1 triangle area 0.5x6x4 (= 12), doubled (= 24)
(b) M1 three rectangles: 6x12, 5x12, 5x12 (= 72, 60, 60) summed
(b) A1 216 cm² cao
(b) Answer: 216 cm²

Question 10

M1 2 x π x 6² (= 72pi or 226.19) seen
M1 2 x π x 6 x 15 (= 180pi or 565.49) seen
A1 792 cm² cso, from unrounded value 791.68... (or better)
Answer: 792 cm² (given, cso)

Question 11

M1 trapezium area 0.5x(4+12)x3 (= 24), doubled (= 48)
M1 perimeter of trapezium 4+5+12+5 (= 26)
M1 26 x 10 (= 260), or complete method
A1 308 cm² cao
Answer: 308 cm²

Question 12

M1 one pair of walls: 2 x (3x2.4) (= 14.4)
M1 other pair of walls: 2 x (2x2.4) (= 9.6)
M1 total wall area 14.4 + 9.6 (= 24) m²
M1 24 / 5 (= 4.8), rounded up to 5 tins (must buy whole tins)
A1 62.50 pounds cao
Answer: 62.50 pounds

Question 13

M1 cube root of 343 (= 7 cm), or 7³ = 343 seen
M1 6 x 7² (= 6 x 49)
A1 294 cm² cao
Answer: 294 cm²

Question 14

M1 2 x π x 5² (= 50pi or 157.08) seen
M1 534 - 157.08 (= 376.92) oe
M1 376.92 / (2 x π x 5) oe
A1 awrt 12 cm
Answer: awrt 12 cm

Question 15

M1 surface area of the large cube, 6x8² (= 384)
B1 correct reasoning that the two small square faces removed (2x2²) are exactly replaced by two new small square faces exposed (2x2²), so the surface area is unchanged
A1 384 cm² cao
Answer: 384 cm²

Question 16

M1 identifies linear scale factor = 3
M1 area scale factor = 3² (= 9)
M1 150 x 9
A1 1350 cm² cao
Answer: 1350 cm²

Question 17

M1 total surface area of the cuboid without the hole, 2(20x10)+2(20x10)+2(10x10) (= 1000)
M1 area of the two circles removed, 2 x π x 3² (= 56.5)
M1 curved surface area of the hole, 2 x π x 3 x 20 (= 377.0)
M1 complete method, 1000 - 56.5 + 377.0
A1 awrt 1320 cm²
Answer: 1320 cm² (3 s.f.)

Question 18

M1 4 x π x 9² (= 324pi or 1017.9) seen
A1 awrt 1020 cm²
Answer: 1020 cm² (3 s.f.)

Question 19

M1 curved surface area 2 x π x 6² (= 72pi or 226.2) seen
M1 flat circular face π x 6² (= 36pi or 113.1) seen
A1 awrt 339 cm²
Answer: 339 cm² (3 s.f.)

Question 20

(a) M1 √5² + 12² oe
(a) A1 13 cm cao
(a) Answer: 13 cm
(b) M1 π x 5 x 13 (= 65pi or 204.2) seen
(b) M1 π x 5² (= 25pi or 78.5) seen
(b) A1 awrt 283 cm²
(b) Answer: 283 cm² (3 s.f.)

Surface Area - Worksheets, Questions and Revision

4.12 Surface Area

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