Exam-Style Questions on Mensuration
Problems on Mensuration adapted from questions set in previous exams.
A builder needs to lift a steel block. It is a cuboid with dimensions 2 m by 0.2 m by 0.2 m. Steel has a density of 7.6 g/cm3.
The builder's lifting gear can lift a maximum load of 500 kg. Can the lifting gear be used to lift the steel block?
Justify your decision.
[The surface area of a sphere of radius \(r\) is \(4\pi r^2\) and the volume is \(\frac43\pi r^3\)]
A solid metal sphere has a radius of 7.5 cm.
(a) Calculate the volume of the sphere to the nearest cubic centimetre.
(b) Calculate the surface area of the sphere to the nearest square centimetre.
(c) If one cubic centimetre of the metal has a mass of 4.9 grams calculate the mass of the sphere to three significant figures.
(d) Two of these spheres are placed in the water in a cylindrical tank with base diameter 32cm. Before they were lowered in the depth of the water was 19cm. Calculate the new depth of water in the cylinder when the spheres are fully submerged.
A circular dart board has radius of 30 cm.
(a) Calculate the area of the front face of the dart board in cm2, giving your answer as a multiple of \(\pi\).
(b) The volume of the dart board is 4500\(\pi\) cm3. Calculate the thickness of the dart board.
A batch of coins is made from an alloy consisting of 270g of copper mixed with 90g of nickel.
(a) Work out the volume of copper used in the alloy.
(b) What is the density of the alloy to three significant figures?
A solid metal cylinder has a base radius of 5cm and a height of 9cm.
(a) Find the area of the base of the cylinder.
(b) Find the volume of the metal used in the cylinder.
(c) Find the total surface area of the cylinder.
The cylinder was melted and recast into a solid cone with a circular base radius, OB (where O is the centre of the circle), of 7cm. The vertex of the cone is the point C.
(d) Find the height, OC, of the cone.
(e) Find the size of angle BCO.
(f) Find the slant height, CB.
(g) Find the total surface area of the cone.
Twenty four spherical shaped chocolates are arranged in a box in four rows and six columns.
Each chocolate has a radius of 1.2 cm.
(a) Find the volume of one chocolate.
(b) Write down the volume of 24 chocolates.
The 24 chocolates fit perfectly in the box with each chocolate touching the ones around it or the sides of the box.
(c) Calculate the volume of the box.
(d) Calculate the volume of empty space in the box.
(e) What percentage of the contents of the box is chocolate?
The three sides of an equilateral triangle are tangents to a circle of radius \(r\) cm. The sides of the triangle are each 10cm long.
(a) Calculate the value of \(r\).
The second diagram shows a box in the shape of a triangular prism of length 15cm.
The cross section is an equilateral triangle with sides of length 10cm.
(b) Calculate the volume of the box.
The box contains cookies. Each cookie is a cylinder of radius 2.8cm and height 5mm.
(c) Calculate the largest number of cookies that will fit in the box.
(d) Calculate the volume of one cookie in cubic centimetres.
(e) Calculate the percentage of the volume of the box not filled with cookies.
A square has sides of length \(x\) cm.
The equilateral triangle next to it has sides which are each 3cm more than the length of a side of the square.
(a) Find the perimeter of the square if it is equal to the perimeter of the triangle.
The diagram above show the same square and triangle.
The length of the diagonal of the square is \(d\) cm and the height of the triangle is \(h\) cm.
(b) Which has the greater value, \(d\) or \(h\)?
The diagram, drawn to scale, shows a right-angled triangle ABC.
Construct using a ruler and a pair of compasses a rectangle, DEFG, equal in area to the area of the triangle ABC and with DE the same length as AB.
You must show all your construction lines.
Three crayons are held together with an elastic band. The diagram below shows the end of the crayons and the elastic band.
Each of the crayons has diameter of 10 mm. Find the length of the elastic band in this position
(a) Find the area of a regular octagon if the distance from its centre to any vertex is 10cm.
(b) If the octagon had been cut from a piece of square card that was only just large enough, work out the area of the original square piece of card.
(c) A table top is made in the shape of a regular octagon with sides five times as long as the card model. Find the ratio of the area of the table top to the area of the card model.