Perimeter, area and volume – Year 7 maths test answers to download as PDF

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1. Answer to Question 1 (4 marks)

   1. Area of a triangle = \(\dfrac{1}{2}\times\) base \(\times\) perpendicular height \(=\dfrac{1}{2}\times14\times9=7\times9=\)63 cm². (1 mark)
   2. Area = length × width, so the length is \(72\div6=12\) cm (1 mark). The perimeter adds all four sides: \(2(12+6)=2\times18=\)36 cm (1 mark).
   3. Area of a trapezium = half the sum of the parallel sides × height: \(\dfrac{1}{2}(7+13)\times5=\dfrac{1}{2}\times20\times5=\)50 cm². (1 mark)

   Remember that area is measured in square units (cm²) and perimeter in ordinary length units (cm). In (b), check that \(12\times6=72\) before working out the perimeter. The trapezium in (c) has the same area as a rectangle 10 cm by 5 cm.

2. Answer to Question 2 (4 marks)

   1. Volume of a cuboid = length × width × height: \(9\times6\times5=54\times5=\)270 cm³. (1 mark)
   2. A cuboid has three pairs of identical faces: \(9\times6=54\), \(9\times5=45\) and \(6\times5=30\) cm² (1 mark). The surface area is \(2(54+45+30)=2\times129=\)258 cm² (1 mark).
   3. The new base has area \(5\times6=30\) cm². Volume = base area × height, so the height is \(270\div30=\)9 cm. (1 mark)

   Notice that the second cuboid is the first one turned over: 5 cm, 6 cm and 9 cm are the same three dimensions, which is why the volume is unchanged. Check: \(5\times6\times9=270\).

3. Answer to Question 3 (4 marks)

   1. 1 m = 100 cm, so 1 m² = \(100\times100=10\,000\) cm². Then \(2.5\times10000=\)25 000 cm². (1 mark)
   2. 1 litre = 1 000 cm³, so divide by 1 000: \(3500\div1000=\)3.5 litres. (1 mark)
   3. 1 m³ holds exactly 1 000 litres, so \(0.4\times1000=\)400 litres. (1 mark)
   4. Use the same factor as in part (a): \(0.3\times10000=\)3 000 cm². (1 mark)

   The most common mistake is multiplying by 100 for area. A square metre is 100 cm long and 100 cm wide, so the factor for area is 10 000. For volume the factor from m³ to cm³ is \(100^{3}=1\,000\,000\).

4. Answer to Question 4 (4 marks)

   1. The playground is a rectangle: \(40\times25=\)1 000 m². (1 mark)
   2. The sandpit takes up \(6\times4=24\) m², which does not need the safety surface. The area to cover is \(1000-24=\)976 m². (1 mark)
   3. Each square metre costs £12, so the surface costs \(976\times12=9760+1952=\)£11 712. (1 mark)
   4. The fence goes round the outside of the playground: perimeter \(2(40+25)=130\) m. The cost is \(130\times8=\)£1 040. (1 mark)

   The sandpit sits in a corner inside the rectangle, so it does not change the perimeter. Estimate for (c): about 1 000 m² at £12 is about £12 000.

5. Answer to Question 5 (4 marks)

   1. The tank is 45 cm high and the water stops 5 cm below the top, so the water is \(45-5=\)40 cm deep. (1 mark)
   2. The water forms a cuboid 80 cm by 35 cm by 40 cm: \(80\times35\times40=112\,000\) cm³ (1 mark). Since 1 litre = 1 000 cm³, this is \(112000\div1000=\)112 litres (1 mark).
   3. When full the tank holds \(80\times35\times45=126\,000\) cm³ = 126 litres. The bigger tank holds 200 litres, so it is \(200-126=\)74 litres bigger. (1 mark)

   Check (b): the base area is \(80\times35=2\,800\) cm², and each centimetre of depth adds 2.8 litres. Forty centimetres give \(40\times2.8=112\) litres.

Total : 20 marks