Volume and surface area

The diagram shows a sphere with radius r cm.

The sphere has a circumference of 50 cm.

Work out the surface area of the sphere.

[The surface area, A, of a sphere with radius r is A = 4πr².]

The diagram shows a sphere with radius 3 cm and a cube with side length x cm.

The surface area of the sphere is equal to the surface area of the cube.

Work out the side length, x cm, of the cube. You must show your working.

[The surface area, A, of a sphere with radius r is A = 4πr².]

The diagram shows the cross-section of a triangular prism.

Work out the area of the cross-section.

This diagram shows the triangular prism.

Work out the total surface area of the triangular prism. You must show your working.

Two of these triangular prisms are joined to make a new prism.

Give a reason why the total surface area of this prism is not two times your answer in part (b).

A cylinder has a radius of 8.4 cm. The ratio of the radius of the cylinder to the height of the cylinder is 2: 5.

Find the volume of the cylinder.

Write down the mathematical name of this solid. Choose from the list in the box.

The diagram shows a cuboid.

How many edges does the cuboid have?

The cuboid has dimensions 10 cm, 7 cm and 5 cm.

Work out the volume of the cuboid.

The diagram shows a regular pentagon made using ten congruent right-angled triangles. The length of one side of the pentagon is 12 cm.

Show that the area of the pentagon is 247.75 cm², correct to 2 decimal places.

The regular pentagon is the base of a pyramid.

The pyramid has volume 450 cm³. The perpendicular height of the pyramid is h cm.

Calculate the value of h. [The volume of a pyramid is × area of base × perpendicular height.]

A sphere has radius x cm. A cone has radius R cm and height 2 R cm.

The volume of the sphere is equal to the volume of the cone.

Write R in terms of x.

The diagram shows a regular pentagon made using ten congruent right-angled triangles. The length of one side of the pentagon is 12 cm.

Show that the area of the pentagon is 247.75 cm², correct to 2 decimal places.

The regular pentagon is the base of a pyramid.

The pyramid has volume 450 cm³. The perpendicular height of the pyramid is h cm.

Calculate the value of h. [The volume of a pyramid is × area of base × perpendicular height.]

The diagram shows a cuboid with dimensions 12 cm, 5 cm and x cm.

The volume of the cuboid is 240 cm³.

Find the value of x.

The diagram shows a cone and a cube. The cone has radius 8 cm and height 10 cm.

The volume of the cone is equal to the volume of the cube.

Work out the length of one side of the cube. [The volume V of a cone with radius r and height h is V = r²h.]

Work out the surface area of the cuboid.

A sphere has a radius of 3 cm.

Calculate the volume of the sphere. Give your answer in terms of π in its simplest form. [The volume V of a sphere with radius r is V = r³.]

The diagram shows a prism of length 10 cm.

The cross-section of the prism is a right-angled triangle. The base, b cm, is 2 cm longer than the height, h cm. The volume of the prism is 240 cm³.

A student is explaining how they worked out the value of b.

They say

b is 6 because that means h is 4 and 6 × 4 × 10 = 240.

Describe the student’s error and find the correct value of b.

A block made of iron is in the shape of a cuboid. The block is 3.1 cm by 4.9 cm by 2.2 cm. The density of iron is 7.87 g / cm³. Sam thinks that the mass of the block is about 2.4 kg.

Use estimation to decide if Sam’s answer is reasonable. Show how you decide.

Sam’s answer is........ because........

The diagram shows a square-based pyramid ABCDE. O is the centre of the base.

The pyramid has base length 20 cm and each sloping edge has length 14.5 cm.

Draw the plan view of the pyramid on the one-centimetre grid below.

Scale: 1 cm represents 4 cm.

Calculate the volume of the pyramid. You must show your working. [The volume of a pyramid is × area of base × perpendicular height]

A child has four identical wooden cubes of side length 6 cm.

They arrange the cubes in a 2 by 2 by 1 arrangement to form a cuboid.

Show that the surface area of the cuboid is 576 cm².

The child rearranges the cubes in a 4 by 1 by 1 arrangement to form a different cuboid.

Calculate the percentage increase in surface area for this cuboid compared with the 2 by 2 by 1 cuboid.

A solid block of wood is a cuboid which measures 3 cm by 4 cm by 5 cm. Its density is 0.65 g/cm³.

Work out the mass of the block of wood.

Here are two areas.

State which area is greater. Show how you decide.

Area........ is greater because........

The diagram shows a sphere and a cone.

The sphere has radius 4 cm. The cone has radius 15 cm and height 30 cm.

The sphere is completely filled with water. The same amount of water is poured into the cone.

Work out the depth, d cm, of the water in the cone. You must show your working. [The volume V of a sphere with radius r is V = r³. The volume V of a cone with radius r and height h is V = r²h.]

A child has four identical wooden cubes of side length 6 cm.

They arrange the cubes in a 2 by 2 by 1 arrangement to form a cuboid.

Show that the surface area of the cuboid is 576 cm².

The child rearranges the cubes in a 4 by 1 by 1 arrangement to form a different cuboid.

Calculate the percentage increase in surface area for this cuboid compared with the 2 by 2 by 1 cuboid.

The diagram shows a square-based pyramid and a sphere.

The pyramid has base length 12.3 cm and perpendicular height 15.7 cm. The sphere has radius r cm.

The pyramid and the sphere have the same volume.

Work out the radius of the sphere. You must show your working. [The volume of a pyramid is × area of base × perpendicular height. The volume V of a sphere with radius r is V = r³.]

The diagram shows a triangular prism and a cube. The ends of the prism are right-angled triangles with base 16 cm and height 12 cm. The prism is 18 cm long.

The volume of the prism is equal to the volume of the cube.

Find the surface area of the cube. You must show your working.

The diagram shows a square-based pyramid and a sphere.

The pyramid has base length 12.3 cm and perpendicular height 15.7 cm. The sphere has radius r cm.

The pyramid and the sphere have the same volume.

Work out the radius of the sphere. You must show your working. [The volume of a pyramid is × area of base × perpendicular height. The volume V of a sphere with radius r is V = r³.]

The diagram shows a triangular prism and a cube. The ends of the prism are right-angled triangles with base 16 cm and height 12 cm. The prism is 18 cm long.

The volume of the prism is equal to the volume of the cube.

Find the surface area of the cube. Y ou must show your working.

The diagram shows a prism.

Draw an accurate plan view of the prism on the one-centimetre square grid below.

Show that the volume of the prism is 72 cm³.

A cuboid with a square base also has a volume of 72 cm³. The height of the cuboid is 2 cm.

Work out the length of one side of the square base.

A student has some cubes that are all the same size. Each cube is 3 cm by 3 cm by 3 cm.

They put 4 of these cubes together to make this shape.

Calculate the surface area of the shape.

Work out the volume of this prism.

A solid metal sphere has mass 235 g. The density of the metal is 7.78 g/cm³.

Show that the surface area of this sphere is 46.9 cm², correct to 3 significant figures. You must show your working. [For a sphere with radius r: Volume = r³ Surface area = 4π r².]

The diagram shows a pyramid ABCDE.

The pyramid has a square horizontal base ABCD with side 5.6 cm.

The vertex E is vertically above the centre O of the base. The height OE of the pyramid is 6.8 cm.

Calculate the surface area of the pyramid. You must show your working.

Work out the volume of this cuboid.

A truck is used to transport some wood panels. Each wood panel is a cuboid measuring 2.4 m by 1.2 m by 1.8 cm. The density of each wood panel is 750 kg/m³.

The truck can carry 15 tonnes of these wood panels.

Calculate the maximum number of wood panels that the truck can carry. Show how you decide.

A cuboid measures 6 cm by 8 cm by 15 cm. A cube has the same volume as the cuboid.

Find the surface area of the cube, giving your answer correct to 3 significant figures.

A truck is used to transport some wood panels. Each wood panel is a cuboid measuring 2.4 m by 1.2 m by 1.8 cm. The density of each wood panel is 750 kg/m³.

The truck can carry 15 tonnes of these wood panels.

Calculate the maximum number of wood panels that the truck can carry. Show how you decide.

The base of a cone is fixed to the top of a cylinder to make a decoration.

The radius of the base of the cone and of the cylinder is r cm. The cone’s height is 5 r cm. The total height of the decoration is 6 r cm. The total volume of the decoration is 225 cm³.

Calculate the value of r. Show your working.

The volume of a cube is 125 cm³.

Calculate the total surface area of the cube. Give the units of your answer.

The diagram shows a right-angled triangular prism ABCDEF.

Length AD = 11 cm, length CD = 10 cm and length CF = 6 cm.

Calculate the volume of the prism.

Use trigonometry to show that angle FDC = 31°, correct to the nearest degree.

Calculate the exact length of AF.

An octahedron is formed from two identical square based pyramids. The square bases are stuck together as shown.

The volume of the octahedron is 60 cm³. The length of the side of each pyramid’s square base is 5 cm.

Work out the height h cm of the octahedron. [The volume of a pyramid is × area of base × perpendicular height]

The length of the longest diagonal of a cube is 25 cm.

Calculate the total surface area of the cube.

A cone has radius r cm and height h cm.

The height is three times the radius. The volume of the cone is 2100 cm³.

Calculate the radius of the cone.

The diagram shows a cylinder and a sphere.

The cylinder has radius 12 cm and height 30 cm. The cylinder and the sphere have the same volume.

Work out the radius r cm of the sphere.

The diagram shows a cylinder and a cone.

The cylinder has radius 2 cm and height 9 cm. The cone has radius r cm and height h cm.

The ratio r: h is 1: 4. The volume of the cone is equal to the volume of the cylinder.

Work out the value of r.

Calculate the volume of a sphere with radius 6 cm.

An ornament is made from a solid glass square-based pyramid. The base has side length 15 cm. A hemisphere with radius 6 cm is cut out of the base of the pyramid. This reduces the volume of glass contained in the ornament by 30%.

Calculate the perpendicular height of the pyramid. [The volume of a pyramid is × area of base × perpendicular height. A hemisphere is half a sphere.]

Calculate the volume of this cuboid.

In this cuboid all lengths are in centimetres.

The cuboid has a volume of 320 cm³.

Find the value of k.

The following formula is for the area, A, of the curved surface area of a cone. A = π rl, where r is the radius and l is the slant height of the cone.

Calculate the total surface area of a cone with radius 5 cm and slant height 12 cm.

The front and side elevations of a prism, with a pentagon as its cross section, are drawn on this one-centimetre square grid.

Draw accurately the plan of the prism on the grid below.

Calculate the volume of the prism.

A sculpture is formed from a cylinder resting on top of a cuboid. The cylinder has radius 45 cm and height 80 cm. The cuboid measures 90 cm by 90 cm by 150 cm.

The sculpture is made of granite. The granite has a density of 2.7 g/cm³.

Calculate the total mass of the sculpture in tonnes.

The object below is made from a square-based pyramid joined to a cuboid.

The base of the cuboid and the base of the pyramid are both squares of side 4 cm. The height of the cuboid is 8 cm and the total height of the object is 13 cm. The total mass of the object is 158 g.

The cuboid is made from wood with density 0.67 g/cm³. The pyramid is made from granite.

Calculate the density of the granite. [The volume of a pyramid is _× area of base_× perpendicular height.]