MensurationCircles, arcs and sectors
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6 Calculate the area of a circle with radius 5.1 cm. .......................................cm2 [2]
IGCSE Mathematics 0580 · Topic practice
Practise perimeter, area, surface area and volume, including arcs, sectors and compound shapes. Keep units visible and convert them before you substitute into a formula.
Appears inPaper 1 78Paper 2 160Paper 3 64Paper 4 184
You’re practising 2016–2018 papers. The 2019–2026 papers, including 2026, need a plan.
Unlock from $6/mo6 Calculate the area of a circle with radius 5.1 cm. .......................................cm2 [2]
10 A water tank in the shape of a cuboid has length 1.5 metres and width 1 metre. The water in the tank is 60 centimetres deep. Calculate the number of litres of water in the tank. ....................................... litres [3]
14 7 cm 1 cm NOT TO SCALE The diagram shows a solid cuboid with base area 7 cm2. The volume of this cuboid is 21 cm3. Work out the total surface area. ......................................... cm2 [3]
15 Find the volume of a cylinder of radius 5 cm and height 8 cm. Give the units of your answer. .............................. ................ [3]
15 A car travels at 108 km/h for 20 seconds. Calculate the distance the car travels. Give your answer in metres. ............................................ m [3]
16 (a) The length of the side of a square is 12 cm, correct to the nearest centimetre. Calculate the upper bound for the perimeter of the square. ........................................... cm [2] (b) Jo measures the length of a rope and records her measurement correct to the nearest ten centimetres. The upper bound for her measurement is 12.35 m. Write down the measurement she records. ............................................. m [1]
16 1.6 m 2.8 m A B C NOT TO SCALE (a) Find the area of triangle ABC. ............................................. m2 [2] (b) Calculate AC. AC = .................................... m [2]
19 72° 6 cm NOT TO SCALE The diagram shows a sector of a circle with radius 6 cm and sector angle 72°. The perimeter of this sector is ( ) p qr + cm. Find the value of p and the value of q. p = ................................................ q = ................................................ [3]
21 M N NOT TO SCALE A 10 cm C B The diagram shows an equilateral triangle ABC with sides of length 10 cm. AMN is a sector of a circle, centre A. M is the mid-point of AC. Work out the area of the shaded region. ......................................... cm2 [4]
2 D NOT TO SCALE C A B The vertices of a square ABCD lie on the circumference of a circle, radius 8 cm. (a) Calculate the area of the square. ......................................... cm2 [2] (b) (i) Calculate the area of the shaded segment. ......................................... cm2 [3] (ii) Calculate the perimeter of the shaded segment. .......................................... cm [4]
3 (a) NOT TO SCALE 8 cm 17 cm The diagram shows a solid cone. The radius is 8 cm and the slant height is 17 cm. (i) Calculate the curved surface area of the cone. [The curved surface area, A, of a cone with radius r and slant height l is A rl r = .] ........................................... cm2 [2] (ii) Calculate the volume of the cone. [The volume, V, of a cone with radius r and height h is V r h 3 1 2 r = .] ........................................... cm3 [4] (iii) The cone is made of wood and 1 cm3 of the wood has a mass of 0.8 g. Calculate the mass of the cone. ............................................... g [1] (iv) The cone is placed in a box. The total mass of the cone and the box is 1.2 kg. Calculate the mass of the box. Give your answer in grams. ............................................... g [1] 7 (b) NOT TO SCALE 3r 8r r The diagram shows a solid cylinder and a solid sphere. The cylinder has radius 3r and height 8r. The sphere has radius r. (i) Find the volume of the sphere as a fraction of the volume of the cylinder. Give your answer in its lowest terms. [The volume, V, of a sphere with radius r is V r 3 4 3 r = .] .................................................. [4] (ii) The surface area of the sphere is 81r cm2. Find the curved surface area of the cylinder. Give your answer in terms of r. [The surface area, A, of a sphere with radius r is A r 4 2 r = .] ........................................... cm2 [4]
4 (a) Simplify. (i) (3p2)5 ................................................. [2] (ii) 18x2y6 ' 2xy2 ................................................. [2] (iii) m 5 2 - c m ................................................. [1] (b) In this part, all measurements are in metres. 5x – 9 NOT TO SCALE 3x + 7 w The diagram shows a rectangle. The area of the rectangle is 310 m2. Work out the value of w. w = ................................................ [4]
5 NOT TO SCALE O B A C 8 cm 7 cm 78° The diagram shows a design made from a triangle AOC joined to a sector OCB. AC = 8 cm, OB = OC = 7 cm and angle ACO = 78°. (a) Use the cosine rule to show that OA = 9.47 cm, correct to 2 decimal places. [4] (b) Calculate angle OAC. Angle OAC = ................................................ [3] 9 (c) The perimeter of the design is 29.5 cm. Show that angle COB = 41.2°, correct to 1 decimal place. [5] (d) Calculate the total area of the design. ......................................... cm2 [4]
10 5 (a) At a football match, the price of an adult ticket is $x and the price of a child ticket is $ . x 2 50 - ^ h. There are 18 500 adults and 2400 children attending the football match. The total amount paid for the tickets is $320 040. Find the price of an adult ticket. $ ................................................ [4] (b) (i) Factorise y y 5 84 2 + - . ................................................ [2] (ii) y cm NOT TO SCALE (y + 5) cm The area of the rectangle is 84 cm2. Find the perimeter. .......................................... cm [3] 11 (c) In a shop, the price of a monthly magazine is $m and the price of a weekly magazine is $ . m 0 75 - ^ h. One day, the shop receives • $168 from selling monthly magazines • $207 from selling weekly magazines. The total number of these magazines sold during this day is 100. (i) Show that m m 50 225 63 0 2 - + = . [3] (ii) Find the price of a monthly magazine. Show all your working. $ ............................................... [3]
5 (a) 15.2 cm 7 cm NOT TO SCALE The diagram shows a solid prism with length 15.2 cm. The cross-section of this prism is a regular hexagon with side 7 cm. (i) Calculate the volume of the prism. ......................................... cm3 [5] (ii) Calculate the total surface area of the prism. ......................................... cm2 [3] (b) Another solid metal prism with volume 500 cm3 is melted and made into 6 identical spheres. Calculate the radius of each sphere. [The volume, V, of a sphere with radius r is V r 3 4 3 r = .] ........................................... cm [3]
5 6 cm 18 cm h cm x° NOT TO SCALE The diagram shows a prism with length 18 cm and volume 253.8 cm3. The cross-section of the prism is a right-angled triangle with base 6 cm and height h cm. (a) (i) Show that the value of h is 4.7 . [3] (ii) Calculate the value of x. x = ............................................... [2] (b) Calculate the total surface area of the prism. ........................................ cm2 [6]
6 A solid hemisphere has volume 230 cm3. (a) Calculate the radius of the hemisphere. [The volume, V, of a sphere with radius r is V r 3 4 3 r = .] .......................................... cm [3] (b) A solid cylinder with radius 1.6 cm is attached to the hemisphere to make a toy. NOT TO SCALE The total volume of the toy is 300 cm3. (i) Calculate the height of the cylinder. .......................................... cm [3] 9 (ii) A mathematically similar toy has volume 19 200 cm3. Calculate the radius of the cylinder for this toy. .......................................... cm [3]
6 (a) A D C B E 6 cm 6 cm 12 cm NOT TO SCALE In the pentagon ABCDE, angle ACB = angle AED = 90°. Triangle ACD is equilateral with side length 12 cm. DE = BC = 6 cm. (i) Calculate angle BAE. Angle BAE = ............................................... [4] (ii) Calculate AB. AB = ......................................... cm [2] (iii) Calculate AE. AE = ......................................... cm [3] 13 (iv) Calculate the area of the pentagon. ......................................... cm2 [4] (b) P A B C D S R Q 5 cm 8 cm 4 cm NOT TO SCALE The diagram shows a cuboid. AB = 8 cm, BC = 4 cm and CR = 5 cm. (i) Write down the number of planes of symmetry of this cuboid. ................................................ [1] (ii) Calculate the angle between the diagonal AR and the plane BCRQ. ................................................ [4]
6 B C D A NOT TO SCALE 80° 13 cm 11 cm 4 cm 8 cm (a) Calculate angle ACB. Angle ACB = ................................................. [4] (b) Calculate angle ACD. Angle ACD = .................................................. [4] 13 (c) Calculate the area of the quadrilateral ABCD. ........................................... cm2 [3]
7 In this question, all measurements are in metres. 2x – 3 x 6 NOT TO SCALE The diagram shows a right-angled triangle. (a) Show that 5x2 - 12x - 27 = 0. [3] (b) Solve 5x2 - 12x - 27 = 0. Show all your working and give your answers correct to 2 decimal places. x = ......................... or x = ......................... [4] (c) Calculate the perimeter of the triangle. ............................................ m [2] (d) Calculate the smallest angle of the triangle. ................................................. [2]