Circles, cylinders and cones · Years 8 and 10
In Circles, cylinders and cones (Years 8 and 10) you practise 5 skills: The circumference of a circle · The area of a circle · The volume of a cylinder · The curved surface and surface area of a cylinder · The volume of a cone. Pi (π) is the ratio of the circumference of a circle to its diameter, about 3.14. The circumference of a circle is π times the diameter, and the area of a circle is π times the radius squared.
To keep an answer exact, you write it in terms of π: a circle with a radius of 3 cm has an area of 9π cm². When the question says to use π ≈ 3.14, you multiply by 3.14 and get 28.26 cm². The volume of a cylinder is the area of the base times the height, just as for a cuboid. If you unroll the curved surface of a cylinder, you get a rectangle: its length is the circumference of the base, and its width is the height. The volume of a cone is a third of the volume of the cylinder with the same base and the same height: three of these cones fill the cylinder exactly.
Tip: Check whether you are given the radius or the diameter. The area of a circle and the volume need the radius.
What you learn
The circumference of a circle
For example: The radius of the circle is 2 cm. What is the circumference of the circle? Give your answer in terms of π .
Answer and solution
The answer: 4π
The diameter: 2 × 2 = 4 The circumference of a circle is π times the diameter. π × 4 = 4π
The area of a circle
For example: The radius of the circle is 3 cm. What is the area of the circle? Give your answer in terms of π .
Answer and solution
The answer: 9π
The area of a circle is π times the radius squared. π × 3² = 9π The area of the circle is 9π cm².
The volume of a cylinder
For example: A cylinder has a radius of 1 cm and a height of 10 cm. What is the volume of the cylinder? Give your answer in terms of π .
Answer and solution
The answer: 10π
The area of the base is π × 1² = π , and times the height: π × 10 = 10π . The volume of the cylinder is 10π cm³.
The curved surface and surface area of a cylinder
For example: A cylinder has a radius of 1 cm and a height of 10 cm. What is its curved surface area? Give your answer in terms of π .
Answer and solution
The answer: 20π
The circumference of the base: 2 × π × 1 = 2π The curved surface area is the circumference of the base times the height: 2π × 10 = 20π The curved surface area is 20π cm².
The volume of a cone
For example: The cone has a radius of 1 cm and a height of 15 cm. What is the volume of the cone? Give your answer in terms of π .
Answer and solution
The answer: 5π
The area of the base: π × 1² = π The volume of a cylinder with the same base and the same height: π × 15 = 15π The volume of the cone is a third of it: 15π ÷ 3 = 5π The volume of the cone is 5π cm³.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).