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13+ · Mathematics

CE.M5 Geometry and Measures

ISEB CE CE.M5 · Calculator allowed · about 90 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Convert 3.5 metres into centimetres.

(a)Convert 3.5 m into cm.(1)

(Total for Question 1 is 1 mark)

A rectangle has length 8.6 cm and width 5.4 cm.

(a)Calculate the perimeter of the rectangle.(2)

(Total for Question 2 is 2 marks)

A triangle has a base of 12 cm and a perpendicular height of 7 cm.

(a)Calculate the area of the triangle.(2)

(Total for Question 3 is 2 marks)

A straight line passes through a point. Three angles are formed on one side of the line: 2x, 3x and 40 degrees, in that order.

(a)Find the value of x.(2)

(b)State the size of the largest of the three angles.(1)

(Total for Question 4 is 3 marks)

Two parallel lines are cut by a transversal. Angle a = 118 degrees. Angle b is in the corresponding position at the other intersection. Angle c is co-interior (allied) with angle a.

(a)Find angle b, giving a reason for your answer.(1)

(b)Find angle c, giving a reason for your answer.(2)

(Total for Question 5 is 3 marks)

The diagram shows a regular hexagon.

(a)State the order of rotational symmetry of a regular hexagon.(1)

(b)State the number of lines of symmetry of a regular hexagon.(1)

(Total for Question 6 is 2 marks)

A circle has diameter 21 cm. Use π = 22/7.

(a)Calculate the circumference of the circle.(3)

(Total for Question 7 is 3 marks)

A shape is made from a rectangle 10 cm wide and 6 cm tall with a semicircle of diameter 10 cm on top.

(a)Calculate the total area of the shape, correct to 3 significant figures.(4)

(Total for Question 8 is 4 marks)

A net is made from two congruent regular hexagons and six rectangles, arranged so that it folds up into a closed 3-D solid.

(a)Name the 3-D solid formed by this net.(1)

(b)State the number of faces of this solid.(1)

(c)State the number of vertices of this solid.(1)

(Total for Question 9 is 3 marks)

A cuboid has dimensions 8 cm by 5 cm by 4 cm.

(a)Calculate the volume of the cuboid.(2)

(b)Calculate the total surface area of the cuboid.(2)

(Total for Question 10 is 4 marks)

A cylinder has radius 4 cm and height 10 cm.

(a)Calculate the volume of the cylinder, correct to 3 significant figures.(3)

(b)Give your answer to part (a) in litres, correct to 3 significant figures.(1)

(Total for Question 11 is 4 marks)

A right-angled triangle has legs of 9 cm and 12 cm, with the right angle between them.

(a)Calculate the length of the hypotenuse.(4)

(Total for Question 12 is 4 marks)

A right-angled triangle has hypotenuse 13 cm and one leg of 5 cm.

(a)Calculate the length of the other leg.(4)

(Total for Question 13 is 4 marks)

A map has a scale of 1 : 25000. The distance between the villages of Ashcombe and Bridgeworth on the map is 6.4 cm.

(a)Calculate the real distance between the villages, giving your answer in kilometres.(4)

(Total for Question 14 is 4 marks)

Two triangles are mathematically similar. The smaller triangle has sides of 5 cm and 7 cm. The side on the larger triangle corresponding to the 7 cm side is 10.5 cm. The side on the larger triangle corresponding to the 5 cm side is labelled x.

(a)Find the scale factor from the smaller triangle to the larger triangle.(2)

(b)Calculate the value of x.(2)

(Total for Question 15 is 4 marks)

Triangle A has vertices (1, 1), (3, 1) and (1, 4). Triangle A is translated by the vector (4, -2) to give triangle B. Triangle B is then reflected in the line x = 0 (the y-axis) to give triangle C.

(a)Write down the coordinates of the vertices of triangle B.(2)

(b)Write down the coordinates of the vertices of triangle C.(1)

(Total for Question 16 is 3 marks)

A cyclist travels 45 km in 2 hours 30 minutes at a constant average speed.

(a)Calculate the average speed of the cyclist in km/h.(3)

(b)Convert this speed into metres per second.(2)

(Total for Question 17 is 5 marks)

A metal block has mass 540 g and volume 60 cm^3.

(a)Calculate the density of the metal in g/cm^3.(2)

(b)A second block of the same metal has volume 25 cm^3. Calculate the mass of this block.(2)

(Total for Question 18 is 4 marks)

A pentagon has four interior angles of 100 degrees, 110 degrees, 95 degrees and 130 degrees. The fifth angle is x.

(a)Find the value of x.(4)

(Total for Question 19 is 4 marks)

A ship sails from port A on a bearing of 065 degrees for 10 km to point B. It then changes course and sails on a bearing of 155 degrees for 8 km to point C.

(a)Show that angle ABC is a right angle.(1)

(b)Calculate the direct distance AC, correct to 3 significant figures.(3)

(Total for Question 20 is 4 marks)

In triangle ABC, angle B = 90 degrees, AB = 10 cm and BC = 7 cm.

(a)Calculate the size of angle BAC, correct to 1 decimal place.(4)

(Total for Question 21 is 4 marks)

In triangle PQR, angle Q = 90 degrees, angle P = 42 degrees and the hypotenuse PR = 18 cm.

(a)Calculate the length of QR, correct to 1 decimal place.(3)

(Total for Question 22 is 3 marks)

A garden is trapezium-shaped with parallel sides of 8 m and 14 m, with a perpendicular distance of 6 m between them.

(a)Calculate the area of the garden.(3)

(b)Turf costs 4.50 pounds sterling per square metre. Calculate the total cost of turfing the whole garden.(2)

(Total for Question 23 is 5 marks)

A cylindrical water tank has radius 0.5 m and height 1.2 m. Water flows into the empty tank at a constant rate of 15 litres per minute.

(a)Calculate the volume of the tank, in cubic metres, correct to 3 significant figures.(3)

(b)State the volume of the tank in litres, correct to the nearest litre.(1)

(c)Calculate the time it takes to fill the tank completely, correct to the nearest minute.(2)

(Total for Question 24 is 6 marks)

Two cuboids are mathematically similar. The smaller cuboid has volume 64 cm³ and surface area 96 cm^2. The larger cuboid has volume 216 cm^3.

(a)Find the linear scale factor from the smaller cuboid to the larger cuboid.(3)

(b)Calculate the surface area of the larger cuboid.(3)

(Total for Question 25 is 6 marks)

Mark scheme · CE.M5 Geometry and Measures

Question 1

(a) B1 350 cm cao
(a) Answer: 350 cm

Question 2

(a) M1 8.6 + 5.4 = 14 seen, or all four sides added
(a) A1 28 cm cao
(a) Answer: 28 cm

Question 3

(a) M1 0.5 x 12 x 7 oe seen
(a) A1 42 cm² cao
(a) Answer: 42 cm²

Question 4

(a) M1 2x + 3x + 40 = 180 oe seen
(a) A1 x = 28 cao
(a) Answer: x = 28
(b) B1 84 degrees ft from part a
(b) Answer: 84 degrees

Question 5

(a) B1 118 degrees with reason: corresponding angles are equal
(a) Answer: 118 degrees
(b) M1 180 - 118 oe, using co-interior angles sum to 180 degrees
(b) A1 62 degrees cao
(b) Answer: 62 degrees

Question 6

(a) B1 6 cao
(a) Answer: 6
(b) B1 6 cao
(b) Answer: 6

Question 7

(a) M1 C = π x d oe seen
(a) M1 22/7 x 21 seen
(a) A1 66 cm cao
(a) Answer: 66 cm

Question 8

(a) M1 Rectangle area = 10 x 6 = 60 seen
(a) M1 Semicircle area = 0.5 x π x 5² oe seen
(a) M1 39.26... (or 39.3) evaluated for the semicircle
(a) A1 99.3 cm² awrt cao
(a) Answer: 99.3 cm²

Question 9

(a) B1 hexagonal prism cao
(a) Answer: Hexagonal prism
(b) B1 8 cao
(b) Answer: 8
(c) B1 12 cao
(c) Answer: 12

Question 10

(a) M1 8 x 5 x 4 seen
(a) A1 160 cm³ cao
(a) Answer: 160 cm³
(b) M1 2 x (8x5 + 8x4 + 5x4) oe seen
(b) A1 184 cm² cao
(b) Answer: 184 cm²

Question 11

(a) M1 V = π x r² x h oe seen
(a) M1 π x 4² x 10 seen or evaluated
(a) A1 503 cm³ awrt cao
(a) Answer: 503 cm³
(b) B1 0.503 litres ft from part a
(b) Answer: 0.503 litres

Question 12

(a) M1 c² = a² + b² oe seen
(a) A1 81 + 144 = 225 seen
(a) M1 √225 taken
(a) A1 15 cm cao
(a) Answer: 15 cm

Question 13

(a) M1 b² = c² - a² oe seen
(a) A1 169 - 25 = 144 seen
(a) M1 √144 taken
(a) A1 12 cm cao
(a) Answer: 12 cm

Question 14

(a) M1 6.4 x 25000 oe seen
(a) A1 160000 cm (or 1600 m) seen
(a) M1 convert to km by dividing by 100000 (or 1000, ft from m)
(a) A1 1.6 km cao
(a) Answer: 1.6 km

Question 15

(a) M1 10.5 / 7 oe seen
(a) A1 1.5 (oe 3/2) cao
(a) Answer: 1.5
(b) M1 5 x 1.5 oe (ft from part a)
(b) A1 7.5 cm cao
(b) Answer: 7.5 cm

Question 16

(a) M1 at least one vertex correctly translated by (4, -2)
(a) A1 (5, -1), (7, -1) and (5, 2) all correct cao
(a) Answer: (5, -1), (7, -1), (5, 2)
(b) B1 (-5, -1), (-7, -1) and (-5, 2) all correct ft from part a
(b) Answer: (-5, -1), (-7, -1), (-5, 2)

Question 17

(a) M1 2 hours 30 minutes converted to 2.5 hours
(a) M1 45 / 2.5 oe seen
(a) A1 18 km/h cao
(a) Answer: 18 km/h
(b) M1 18 x 1000 / 3600 oe (ft from part a)
(b) A1 5 m/s cao
(b) Answer: 5 m/s

Question 18

(a) M1 540 / 60 oe seen
(a) A1 9 g/cm³ cao
(a) Answer: 9 g/cm³
(b) M1 9 x 25 oe (ft from part a)
(b) A1 225 g cao
(b) Answer: 225 g

Question 19

(a) B1 sum of interior angles of a pentagon = 540 degrees, using (5-2) x 180
(a) M1 100 + 110 + 95 + 130 = 435 seen
(a) M1 540 - 435 oe seen
(a) A1 105 degrees cao
(a) Answer: 105 degrees

Question 20

(a) B1 cso: 155 - 65 = 90, so angle ABC = 90 degrees
(a) Answer: 90 degrees
(b) M1 AC² = 10² + 8² oe seen (using angle ABC = 90 from part a)
(b) M1 100 + 64 = 164 seen
(b) A1 12.8 km awrt cao
(b) Answer: 12.8 km

Question 21

(a) B1 BC (7 cm) identified as opposite and AB (10 cm) identified as adjacent to angle BAC
(a) M1 tan(BAC) = 7/10 oe seen
(a) M1 BAC = tan^-1(0.7) seen or evaluated
(a) A1 35.0 degrees awrt cao
(a) Answer: 35.0 degrees

Question 22

(a) M1 sin(42) = QR/18 oe seen
(a) M1 QR = 18 x sin(42) evaluated
(a) A1 12.0 cm awrt cao
(a) Answer: 12.0 cm

Question 23

(a) M1 0.5 x (8 + 14) x 6 oe seen
(a) M1 0.5 x 22 x 6 evaluated
(a) A1 66 m² cao
(a) Answer: 66 m²
(b) M1 66 x 4.50 oe (ft from part a)
(b) A1 297 pounds sterling cao
(b) Answer: 297 pounds sterling

Question 24

(a) M1 V = π x r² x h oe seen
(a) M1 π x 0.5² x 1.2 evaluated
(a) A1 0.943 m³ awrt cao
(a) Answer: 0.943 m³
(b) B1 943 litres ft from unrounded value in part a
(b) Answer: 943 litres
(c) M1 942.5 / 15 oe (ft from part b)
(c) A1 63 minutes awrt cao
(c) Answer: 63 minutes

Question 25

(a) M1 volume ratio = 216/64 oe (= 3.375 or 27/8) seen
(a) M1 cube root of the volume ratio taken
(a) A1 1.5 (oe 3/2) cao
(a) Answer: 1.5
(b) M1 area scale factor = 1.5² = 2.25 oe (ft from part a)
(b) M1 96 x 2.25 seen
(b) A1 216 cm² cao
(b) Answer: 216 cm²

Geometry and Measures - Worksheets, Questions and Revision