Area, volume, surface area, arcs and sectors – Year 10 GCSE maths test answers to download as PDF

← Back to the test

1. Answer to Question 1 (4 marks)

   1. \(\tfrac12(8+14)\times6=\) 66 cm² (1 for the formula, 1 for the answer) Add the parallel sides, halve, then multiply by the height: \(\tfrac12\times22\times6=66\).
   2. Circumference \(=2\pi\times7=\) 44.0 cm (1). Area \(=\pi\times49=\) 153.9 cm² (1). The circumference uses \(2\pi r\) and the area uses \(\pi r^2\). So the circumference is 44.0 cm and the area is 153.9 cm².

   In (a), a common slip is to forget the half and give 132 cm². In (b), using \(\pi\approx3.14\) gives 43.96 and 153.86, which round to the same answers; a diameter of 14 cm also gives \(\pi d=44.0\) cm.

2. Answer to Question 2 (4 marks)

   1. Area of triangle \(=\tfrac12\times6\times8=24\) (1). Volume \(=24\times15=\) 360 cm³ (1) A prism's volume is the area of its cross-section times its length. The volume is 360 cm³.
   2. Triangles \(2\times24=48\) (1). Rectangles \(15\times(6+8+10)=360\). Total \(=\) 408 cm² (1) The surface is two triangles and three rectangles, one for each side of the triangle. The total surface area is 408 cm².

   In (b), the three rectangles have different widths (6, 8 and 10 cm), so do not use \(3\times\) one rectangle. Check the units: volume is in cm³ and area in cm².

3. Answer to Question 3 (4 marks)

   1. \(\dfrac{120}{360}\times2\pi\times9=6\pi=\) 18.8 cm (1 for method, 1 for the answer) The sector is \(\tfrac{120}{360}=\tfrac13\) of the circle, and the full circumference is \(18\pi\). The arc length is \(6\pi\) cm, about 18.8 cm.
   2. \(\dfrac{120}{360}\times\pi\times81=27\pi=\) 84.8 cm² (1, 1) Take \(\tfrac13\) of the full area \(81\pi\). The area of the sector is \(27\pi\) cm², about 84.8 cm².

   Exact answers in terms of \(\pi\) are also accepted, \(6\pi\) cm and \(27\pi\) cm². A useful check: arc length times radius divided by 2 gives the sector area, \(6\pi\times9\div2=27\pi\).

4. Answer to Question 4 (4 marks)

   1. Radius \(=0.6\) m. \(V=\pi\times0.36\times2=0.72\pi=2.262\) m³ (1) \(=2262\) litres (1) Halve the diameter to get the radius, use \(V=\pi r^2h\), then multiply by 1000 because 1 m³ is 1000 litres. The tank holds about 2262 litres, as required.
   2. \(2262\times0.68=1538.2\), so £1538 (1 for multiplying, 1 for the answer) Change 68p into £0.68 before multiplying. It costs about £1538 to fill the tank.

5. Answer to Question 5 (4 marks)

   1. \(\sqrt{36+64}=\) 10 cm (1) The radius, the height and the slant height form a right-angled triangle, so the slant height is 10 cm.
   2. \(\tfrac13\pi\times36\times8=\) \(96\pi\) cm³ (1) Use \(V=\tfrac13\pi r^2h\) with \(r=6\) and \(h=8\). The volume is \(96\pi\) cm³.
   3. \(\tfrac43\pi\times27=\) \(36\pi\) cm³ (1) Use \(V=\tfrac43\pi r^3\) with \(r^3=27\). The volume is \(36\pi\) cm³.
   4. \(96\pi:36\pi=96:36=\) 8 : 3 (1) The \(\pi\) cancels, then divide both numbers by their HCF, 12. The ratio is 8 : 3.

Total : 20 marks