Key points
A good understanding of calculations for the circumference of a circle and the area of a circle is useful when calculating the surface area and volume of a cylinderA 3D shape with a constant circular cross-section across its length..
A cylinder is a 3DThree-dimensions: length, width and height. shape with a circular cross-sectionThe face that results from slicing through a solid shape. .
The total surface area (of a 3D shape)The total area of all the faces of a 3D shape. Measured in square units, such as cm虏 and m虏. of a cylinder is made up of two circular faceOne of the flat surfaces of a solid shape. and a curved surface which makes a rectangle if flattened out. Surface area is measured in square units, such as cm虏 and m虏.
The volumeThe amount of space occupied by a 3D shape, measured in cubic units, such as cm鲁, mm鲁 and m鲁. May also be referred to as capacity. of a cylinder is the areaA measure of the size of any plane surface or 2D shape. Area is measured in square units, for example, square centimetres or square metres: cm虏 or m虏. of its cross-section, a circle, multiplied by its height. Volume is measured in cubic units, such as cm鲁 or m鲁.
Calculate the area of a cylinder
A cylinder is made up of two congruentShapes that are the same shape and size, they are identical. circles that are directly opposite one another, and a rectangle.
Calculations may be carried out numerically using a decimal approximation for 蟺 (pi)Pi is used to represent the ratio of a circumference of a circle to its diameter, denoted with the Greek symbol 蟺 (pi), such as 3郯14 or 3郯142. Workings may also be written symbolically in terms of 蟺. This means that the result is given as a multiple of 蟺.
On a scientific calculator, the S
鈬
D button is used to convert a value in terms of 蟺 to a decimal value.
To calculate the surface area of a cylinder:
- Work out the area of the two circular faces (2 脳 蟺\(r\)虏).
- Work out the curved surface area, this is the rectangular face (2蟺\(r\) 脳 \(h\)).
- Sum the area of the circles and the rectangle.
The expression for working out the total surface area of a cylinder is
2蟺\(r\)虏 + 2蟺\(rh\).
\(r\) is the radius of the circular cross-section and \(h\) is the height of the cylinder.
If the diameter, \(d\), of the circular cross-section is given, this is halved to find the radius.
Depending on the orientation of the cylinder, the length of the cylinder is its height.
Example
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Question
Use the formula to work out the surface area of the cylinder.
Use the approximation 蟺 = 3郯14. Give your answer to the nearest mm虏.
The diameter of the circular cross-section is 14 mm.
The radius is half of the diameter. The radius of the circular cross-section is 7 mm.
The length of the cylinder is 16 mm. The length is also the height of the cylinder.
Substitute the values of \(r\) = 7 and \(h\) = 16 into 2蟺\(r\)虏 + 2蟺\(rh\) and work out the calculation.
Surface area = 2 脳 蟺 脳 7虏 + 2 脳 蟺 脳 7 脳 16
Surface area = 2 脳 3郯14 脳 7虏 + 2 脳 3郯14 脳 7 脳 16
Surface area = 1011郯08
The total surface area of the cylinder is 1011 mm虏 to the nearest mm虏.
Calculate the volume of a cylinder
The volume of any cylinder is the area of the cross-section multiplied by its height.
This is given by the formula \(V\) = 蟺\(r\)虏\(h\) , where \(r\) is the radius of the circular cross-section and \(h\) is the height of the cylinder.
To work out the volume of a cylinder:
- Find the area of the cross-section using the formula for the area of a circle, \(A\) = 蟺\(r\)虏.
- Multiply by the height of the cylinder.
Examples
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Question
What is the volume of the cylinder? Use the approximation 蟺 = 3郯14
The diameter of the cylinder is 30 m.
The radius is half of the diameter. The radius of the cylinder is 15 m.
The length of the cylinder is 80 m. The \(h\) in the formula is the cylinder鈥檚 height or length, which means \(h\) = 80
Substitute the values of \(r\), 15, and \(h\), 80, into the formula (\(V\) = 蟺\(r\)虏\(h\)) and work out the calculation.
3郯14 脳 15虏 脳 80 = 56,520
The volume of the cylinder is 56,520 m鲁.
Practise finding the surface area and volume of cylinders
Practise working out the surface area and volume of cylinders with this quiz. You may need a pen and paper to help you with your answers.
Quiz
Real-life maths
Certain foods, such as baked beans, vegetables, fish and meat, are commonly sold in cylinder-shaped tins. Cylinders pack efficiently into boxes for shipping, taking up approximately 90% of the available space.
Their circular cross-section also means that the tins can withstand pressure when stored. The food contained in them has a long shelf life.
To make the tins accurately, manufacturers need to calculate the surface area plus a small amount of extra area for the seams. The volume of the cylinder shape will determine the quantity of food that can go inside a tin.
Game - Divided Islands
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