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Grade 6 · Higher · free GCSE maths worksheet

Volume of cones, spheres and pyramids

Using the formulae provided.

19 original questions, 78 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

A cone has base radius 6 cm and vertical height 10 cm.

Work out the volume of the cone. Give your answer correct to 3 significant figures.

2 marks
Show answer
377 cm³
2

A sphere has radius 3 cm.

  1. Work out the volume of the sphere. Give your answer in terms of π.
  2. Write your answer to part (a) as a number, correct to 3 significant figures.
3 marks
Show answer
a) 36π cm³ b) 113 cm³
3

A pyramid has a square base of side 8 cm. Its perpendicular height is 9 cm.

Work out the volume of the pyramid.

2 marks
Show answer
192 cm³
4

A solid hemisphere has radius 6 cm.

Work out the volume of the hemisphere.

Give your answer correct to 3 significant figures.

2 marks
Show answer
452 cm³
5

A solid is made from a cone and a hemisphere.

The cone has base radius 5 cm and vertical height 9 cm.

The flat face of the hemisphere, of radius 5 cm, is joined to the base of the cone.

Work out the total volume of the solid.

Give your answer correct to 3 significant figures.

5 cm9 cmCentral vertical cross-sectionDiagram NOT accurately drawn
4 marks
Show answer
497 cm³
6

An ice cream cone has base radius 5 cm and volume 250π cm3.

Work out the vertical height of the cone.

3 marks
Show answer
30 cm
7

A solid metal ball has radius 5 cm. The metal has density 7.8 g/cm3.

Work out the mass of the ball in kilograms. Give your answer correct to 3 significant figures.

4 marks
Show answer
4.08 kg
8

A solid is a quarter of a sphere. It is made by cutting a sphere in half, then cutting one hemisphere in half again.

The volume of the solid is 243π cm³.

Work out the total surface area of the solid.

Give your answer in terms of π.

Both cuts pass through the centre of the original sphere, and the second cutting plane is perpendicular to the first.

Diagram NOT accurately drawnSection perpendicular to both cutsDiagram NOT accurately drawnFlat faces, shown separately
5 marks
Show answer
162π cm²
9

A solid cone has base radius 5 cm.

The curved surface area of the cone is 65π cm².

Work out the volume of the cone.

Give your answer correct to 3 significant figures.

5 marks
Show answer
314 cm³
10

A solid hemisphere has volume 18π cm3.

Work out the total surface area of the hemisphere. Give your answer in terms of π.

4 marks
Show answer
27π cm²
11

A solid is made from a cone and a hemisphere, both of radius 4 cm. The flat face of the hemisphere is joined to the base of the cone.

The total volume of the solid is 96π cm³.

Work out the total height of the solid.

4 cmCentral vertical cross-sectionDiagram NOT accurately drawn
5 marks
Show answer
14 cm
12

A solid is made from a cylinder and a cone. Both have radius 4 cm. The cylinder has height 10 cm. The cone sits on top of the cylinder and has vertical height h cm.

The total volume of the solid is 224π cm3.

Work out the value of h.

4 marks
Show answer
12
13

A cube has sides of length 6 cm.

A pyramid has a square base of side 9 cm and height h cm.

Volume of a pyramid = ⅓ × base area × height

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

Work out the value of h.

5 marks
Show answer
8
14

A frustum is made by removing a cone of base radius 3 cm and vertical height 6 cm from a similar cone of base radius 6 cm and vertical height 12 cm.

A solid hemisphere of radius 6 cm is attached to the larger face of the frustum to make a solid.

Calculate the total volume of the solid.

Give your answer correct to 3 significant figures.

6 cm12 cm6 cm3 cmCentral vertical cross-sectionDiagram NOT accurately drawn
5 marks
Show answer
848 cm³
15

A cone has base radius 9 cm and vertical height 15 cm.

A smaller cone of vertical height 5 cm is cut off the top, using a cut parallel to the base. The solid left is a frustum.

Work out the volume of the frustum. Give your answer correct to 3 significant figures.

9 cm5 cm15 cmDiagram NOT accurately drawn
5 marks
Show answer
1230 cm³
16

A solid cone has base radius 6 cm and volume 96π cm³.

The curved surface of the cone is made from a sector of a circle.

Work out the angle of the sector.

5 marks
Show answer
216°
17

A solid sphere has radius r. A solid cone has base radius r and vertical height h.

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

  1. Show that h = 4r.
  2. Find the ratio of the curved surface area of the cone to the surface area of the sphere. Give your answer in the form √k : m.
6 marks
Show answer
a) 1/3 πr²h = 4/3 πr³ gives h = 4r b) √17 : 4
18

A flower pot is in the shape of a frustum. The diagram shows its cross-section. The top is a circle of radius 9 cm, the bottom is a circle of radius 3 cm and the height is 8 cm.

The pot is a frustum, so it is a large cone with a small cone cut off the bottom.

  1. Show that the cone that was cut off has height 4 cm.
  2. Work out the volume of the pot. Give your answer correct to 3 significant figures.
9 cm3 cm8 cmDiagram NOT accurately drawn
4 marks
Show answer
a) Similar triangles: 3/9 = h/(h + 8), so h = 4 b) 980 cm³
19

A toy is made from a solid cylinder of wood with a solid hemisphere of plastic stuck on top. The diagram shows its cross-section. The cylinder and the hemisphere both have radius 4 cm. The cylinder has height 10 cm.

The wood has density 0.6 g/cm3. The plastic has density 1.2 g/cm3.

Work out the average density of the whole toy. Give your answer in g/cm3 correct to 3 significant figures.

4 cm10 cmplasticwoodDiagram NOT accurately drawn
5 marks
Show answer
0.726 g/cm³

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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