Measures, Shape and Space

The end wall of a garden shed is a rectangle 2.4 m wide and 1.8 m high, with a triangle on top for the roof. The triangle has the same 2.4 m base and a height of 0.8 m. What is the area of the end wall?

• A5.28 m²✓
• B10.56 m²
• C4.32 m²
• D6.24 m²

Answer: Rectangle: 2.4 × 1.8 = 4.32 m². Triangle: 1/2 × 2.4 × 0.8 = 0.96 m². Total: 4.32 + 0.96 = 5.28 m². Forgetting to halve the triangle gives 6.24 m², and 10.56 m² is the area of both end walls.

A bicycle wheel has a diameter of 70 cm. About how many turns does the wheel make when the bike travels 1 km, to the nearest whole number? (Use π = 3.14 and 1 km = 100,000 cm.)

• A227
• B455✓
• C1,429
• D910

Answer: One turn moves the bike one circumference: π × d = 3.14 × 70 = 219.8 cm. Number of turns: 100,000 ÷ 219.8 = 454.96..., which is 455 to the nearest whole number. Using π × radius gives 910, and dividing by 70 without π gives 1,429.

A circular patio has a radius of 2.5 m. Paving is sold in whole square metres at £38 per m². Using π = 3.14, how much does the paving cost?

• A£3,002
• B£608
• C£760✓
• D£745.75

Answer: Area = πr² = 3.14 × 2.5 × 2.5 = 19.625 m². Paving is sold in whole square metres, so 20 m² must be bought: 20 × £38 = £760. Not rounding up gives £745.75, and using the circumference (15.7 m) instead of the area gives £608.

A triangular flower bed has a base of 12 m and a perpendicular height of 8 m. Turf costs £4.50 per m². How much does it cost to turf the whole bed?

• A£108
• B£432
• C£216✓
• D£90

Answer: Area of a triangle = 1/2 × base × height = 1/2 × 12 × 8 = 48 m². Cost: 48 × £4.50 = £216. Forgetting to halve gives 96 m² and £432; adding base and height gives 20 m² and £90.

A cylindrical water tank has a radius of 0.5 m and a height of 1.2 m. What is its volume, to 2 decimal places?

• A1.88 m³
• B3.77 m³
• C0.79 m³
• D0.94 m³✓

Answer: The volume of a cylinder is π × radius² × height: π × 0.5² × 1.2 = 0.94 m³.

A cylindrical tin of paint has a diameter of 16 cm and a height of 20 cm. What is its volume, to the nearest cm³?

• A4,021 cm³✓
• B16,085 cm³
• C2,011 cm³
• D1,005 cm³

Answer: The radius is half of the diameter, which is 8 cm, so the volume is π × 8² × 20 = 4,021 cm³.

A hot-water cylinder has a radius of 20 cm and a height of 90 cm. How many litres does it hold, to the nearest litre? (1 litre = 1,000 cm³)

• A452 litres
• B113 litres✓
• C1,131 litres
• D11 litres

Answer: The volume is π × 20² × 90 = 113,097 cm³, and dividing by 1,000 converts that to 113 litres.

A cylindrical bin has a radius of 25 cm and a height of 70 cm. A label wraps right around its curved side. What area of label is needed, to the nearest cm²?

• A5,498 cm²
• B21,991 cm²
• C10,996 cm²✓
• D1,963 cm²

Answer: The curved surface area of a cylinder is 2 × π × radius × height: 2 × π × 25 × 70 = 10,996 cm².

A closed cylindrical tin has a radius of 5 cm and a height of 12 cm. What is its total surface area, to the nearest cm²?

• A377 cm²
• B534 cm²✓
• C456 cm²
• D942 cm²

Answer: The total is the curved side plus both circular ends: (2 × π × 5 × 12) + (2 × π × 5²) = 534 cm².

A cylindrical tank has a base area of 2 m² and holds 5 m³ of water when it is full. How tall is the tank?

• A10 m
• B0.4 m
• C3 m
• D2.5 m✓

Answer: The volume of a cylinder is its base area multiplied by its height, so the height is 5 ÷ 2 = 2.5 m.

Two cylindrical tins are the same height. Tin A has a radius of 4 cm and tin B has a radius of 8 cm. How many times more paint does tin B hold than tin A?

• A2 times
• B8 times
• C4 times✓
• D16 times

Answer: Volume depends on the radius squared, so doubling the radius multiplies the volume by 2 × 2 = 4 times.

A cylindrical boiler is 1.5 m tall and has a diameter of 0.8 m. What area of insulation is needed to wrap around its curved side, to 2 decimal places?

• A3.77 m²✓
• B7.54 m²
• C1.88 m²
• D0.50 m²

Answer: The radius is half of the diameter, which is 0.4 m, so the curved surface area is 2 × π × 0.4 × 1.5 = 3.77 m².

On a site plan, three corners of a rectangular car park are at (-3, 2), (5, 2) and (5, -4). What are the co-ordinates of the fourth corner?

• A(-3, 4)
• B(-3, -4)✓
• C(3, -4)
• D(-4, -3)

Answer: Two corners share y = 2 and two share x = 5. The fourth corner must line up with (-3, 2) vertically and with (5, -4) horizontally, so it takes x = -3 and y = -4: (-3, -4). Writing the numbers the other way round gives (-4, -3), which is a different point.

A delivery drone flies in a straight line from a depot at (-2, 5) to a customer at (6, -1). It passes a checkpoint exactly halfway along its route. What are the co-ordinates of the checkpoint?

• A(4, 4)
• B(-4, 3)
• C(4, -3)
• D(2, 2)✓

Answer: Halfway means the mean of each co-ordinate. x: (-2 + 6) ÷ 2 = 2. y: (5 + -1) ÷ 2 = 2. The checkpoint is at (2, 2). Halving the differences (8 and -6) gives (4, -3), which is the size of half the journey, not a position.

On a town grid, the library is at (-4, 3). The station is 7 units to the right of the library and 5 units down. What are the co-ordinates of the station?

• A(-2, 3)
• B(3, 8)
• C(3, -2)✓
• D(-11, -2)

Answer: Moving right adds to the x co-ordinate: -4 + 7 = 3. Moving down subtracts from the y co-ordinate: 3 - 5 = -2. The station is at (3, -2). Adding 5 to y instead gives (3, 8), which is 5 units up.

A square flower bed is marked on a grid. Two neighbouring corners are at (1, 1) and (1, -3), and the other two corners lie to the left of these. What are the co-ordinates of the other two corners?

• A(-2, 1) and (-2, -3)
• B(-3, 1) and (-3, -3)✓
• C(5, 1) and (5, -3)
• D(-3, -1) and (-3, 3)

Answer: The side from (1, 1) to (1, -3) is 1 - (-3) = 4 units long, so every side is 4 units. Moving 4 units left from x = 1 gives x = -3, so the corners are (-3, 1) and (-3, -3). Moving right instead gives (5, 1) and (5, -3).

On a map grid, a lighthouse is at (-6, 4) and a harbour is at (2, 4). Each grid unit represents 0.5 km. How far is the lighthouse from the harbour?

• A4 km✓
• B8 km
• C16 km
• D2 km

Answer: Both places have y = 4, so the distance is the change in x: 2 - (-6) = 8 grid units. Each unit is 0.5 km, so the distance is 8 × 0.5 = 4 km. 8 km forgets to convert the grid units, and 16 km divides by 0.5 instead of multiplying.

A triangle is drawn on a grid with corners at A(-4, 0), B(4, 0) and C(0, 6). What is the area of triangle ABC in square units?

• A12
• B48
• C24✓
• D14

Answer: The base AB runs from x = -4 to x = 4, so it is 8 units long. C is 6 units above the x-axis, so the height is 6. Area = 1/2 × 8 × 6 = 24 square units. Forgetting to halve gives 48.

A robot vacuum starts at (0, 0). It moves 3 units left, then 5 units down, then 8 units right. Where does it finish?

• A(5, -5)✓
• B(-11, -5)
• C(5, 5)
• D(-5, 5)

Answer: Track x and y separately. x: 0 - 3 + 8 = 5. y: 0 - 5 = -5. The robot finishes at (5, -5). Treating 'right' as negative gives x = -11.

A closed cardboard box is a cuboid 40 cm long, 25 cm wide and 15 cm tall, standing on a table. What does its plan (the view from directly above) look like?

• AA rectangle measuring 40 cm by 25 cm✓
• BA square measuring 25 cm by 25 cm
• CA rectangle measuring 25 cm by 15 cm
• DA rectangle measuring 40 cm by 15 cm

Answer: Looking straight down, you see the top of the box, which has the box's length and width: a 40 cm by 25 cm rectangle. The height does not show in the plan. 40 cm by 15 cm is the front elevation, and 25 cm by 15 cm is a side elevation.

A drinks can stands upright on a counter. Which pair correctly describes its plan and its front elevation?

• APlan: a rectangle; front elevation: a circle
• BPlan: a rectangle; front elevation: a rectangle
• CPlan: a circle; front elevation: a rectangle✓
• DPlan: a circle; front elevation: a circle

Answer: From directly above, you see the circular top of the can, so the plan is a circle. From the front, you see its height and width, which make a rectangle. This pair of views is typical of a cylinder standing on one end.

A 3-D solid has a plan that is a square with both diagonals drawn in, and its front and side elevations are both triangles. What is the solid?

• AA cube resting on one of its faces
• BA triangular prism lying on one face
• CA square-based pyramid standing on its base✓
• DA cone standing on its circular base

Answer: The square plan shows a square base, and the diagonals are the sloping edges meeting at the apex above the centre. Triangles from the front and the side confirm the faces slope up to a point, so it is a square-based pyramid. A cone would have a circular plan.

A ridge tent is shaped like a triangular prism. Its front elevation, looking at the entrance, is a triangle 1.6 m wide at the base and 1.2 m high. The tent is 2.5 m long. What is its plan (the view from above)?

• AA rectangle 2.5 m by 1.6 m✓
• BA square with sides of 2.5 m
• CA rectangle 2.5 m by 1.2 m
• DA triangle 1.6 m wide and 1.2 m high

Answer: From above you see the length of the tent (2.5 m) and its width at the ground (1.6 m), so the plan is a 2.5 m by 1.6 m rectangle, with a line along the ridge. The height of 1.2 m cannot be seen from above. The triangle is the front elevation.

An architect's plan of a kitchen uses a scale of 1:50. On the plan, the kitchen floor is a rectangle 9 cm by 7 cm. What is the real area of the kitchen floor?

• A16 m²
• B157.5 m²
• C0.315 m²
• D15.75 m²✓

Answer: Scale each length first: 9 cm × 50 = 450 cm = 4.5 m and 7 cm × 50 = 350 cm = 3.5 m. Real area: 4.5 × 3.5 = 15.75 m². Multiplying the plan area (63 cm²) by 50 only once gives 3,150 cm², or 0.315 m²; 16 m² is the perimeter in metres.

On a set of building drawings, which view of an object is called its side elevation?

• AThe view from directly below, looking up
• BThe view from one side, looking horizontally✓
• CThe view from directly above, looking down
• DThe flat shape that folds up to make it

Answer: An elevation is a view looking horizontally at an object. The side elevation is that view from one side, and the front elevation is from the front. The view from above is the plan, and the flat shape that folds up to make a solid is its net.

A road sign in France shows that Paris is 120 km away. Using 5 miles ≈ 8 km, about how many miles away is Paris?

• A192 miles
• B24 miles
• C96 miles
• D75 miles✓

Answer: 8 km is about 5 miles, so 1 km is 5/8 of a mile. 120 × 5 ÷ 8 = 75 miles. Multiplying by 8/5 instead gives 192, which would make the distance in miles bigger than in kilometres; a mile is longer than a kilometre, so the number of miles must be smaller.

A baker needs 12 ounces of flour per loaf and is baking 5 loaves. Flour is sold in 1.5 kg bags. Using 1 ounce ≈ 28 g, how many bags must she buy?

• A5 bags
• B1 bag
• C19 bags
• D2 bags✓

Answer: Flour needed: 12 × 5 = 60 ounces. 60 × 28 = 1,680 g = 1.68 kg. One 1.5 kg bag is not enough, so she must buy 2 bags. 19 bags comes from using the pound (about 454 g) instead of the ounce.

A car's empty fuel tank holds 11 gallons. Using 1 gallon ≈ 4.5 litres, and with fuel at £1.42 per litre, how much does it cost to fill the tank?

• A£3.47
• B£15.62
• C£70.29✓
• D£49.50

Answer: Convert first: 11 × 4.5 = 49.5 litres. Cost: 49.5 × £1.42 = £70.29. £15.62 prices the gallons as if they were litres, and £49.50 is the number of litres written as money.

A conversion graph between kilograms and pounds is a straight line through (0 kg, 0 lb) and (10 kg, 22 lb). What is 35 kg in pounds?

• A57 lb
• B15.9 lb
• C7.7 lb
• D77 lb✓

Answer: The graph shows 10 kg = 22 lb, so 1 kg = 2.2 lb. 35 × 2.2 = 77 lb. Dividing by 2.2 gives 15.9, which converts the wrong way; a pound is lighter than a kilogram, so there are more pounds than kilograms.

A room measures 15 feet by 12 feet. Carpet is sold by the square metre. Using 1 foot ≈ 0.3 m, what is the area of the room in square metres?

• A8.1 m²
• B54 m²
• C16.2 m²✓
• D600 m²

Answer: Convert each length first: 15 × 0.3 = 4.5 m and 12 × 0.3 = 3.6 m. Area: 4.5 × 3.6 = 16.2 m². Multiplying 180 square feet by 0.3 only once gives 54, because a square foot is 0.3 × 0.3 = 0.09 m², not 0.3 m².

A baby weighs 7 lb 8 oz at birth. There are 16 ounces in a pound, and 1 kg ≈ 2.2 lb. What is the baby's weight in kilograms, to 1 decimal place?

• A17.2 kg
• B3.5 kg
• C16.5 kg
• D3.4 kg✓

Answer: 8 oz is half a pound, so the baby weighs 7.5 lb. 7.5 ÷ 2.2 = 3.409..., which is 3.4 kg. Reading 7 lb 8 oz as 7.8 lb gives 3.5 kg, and multiplying by 2.2 gives 16.5 kg.

A fence panel is 6 feet long. There are 12 inches in a foot, and 1 inch = 2.54 cm. How long is the panel in metres, to 2 decimal places?

• A0.15 m
• B1.52 m
• C18.29 m
• D1.83 m✓

Answer: 6 feet = 6 × 12 = 72 inches. 72 × 2.54 = 182.88 cm, which is 1.8288 m, or 1.83 m to 2 decimal places. Using 10 inches in a foot gives 1.52 m, and 0.15 m converts 6 inches rather than 6 feet.

A paving slab is shaped like a regular hexagon. What is the size of each interior angle of the slab?

• A60°
• B120°✓
• C135°
• D108°

Answer: A hexagon can be split into 4 triangles from one corner, so its angles total 4 × 180° = 720°. A regular hexagon has 6 equal angles: 720° ÷ 6 = 120°. 108° is the angle in a regular pentagon and 135° in a regular octagon.

A closed box is a cuboid 30 cm long, 20 cm wide and 10 cm high. How much wrapping paper is needed to cover the outside exactly, with no overlap?

• A1,100 cm²
• B6,000 cm²
• C3,300 cm²
• D2,200 cm²✓

Answer: A cuboid has three pairs of faces: 30 × 20 = 600, 30 × 10 = 300 and 20 × 10 = 200 cm². Total surface area = 2 × (600 + 300 + 200) = 2 × 1,100 = 2,200 cm². 1,100 cm² counts each pair once, and 6,000 is the volume in cm³.