Cylinders
Volume
A cylinder is a 3D shape with a circular cross section and a curved surface. It has no straight edges. To find the volume or surface area of a cylinder, calculate it in the same as a prism.
For a cylinder of height \(h\) with radius \(r\), the volume of a cylinder can be calculated using the formula:
\(\text{volume of a cylinder} = \pi r^2 h\)
Example
Calculate the volume of the cylinder.
The measurement shown is the diameter of the circular face, so the radius is 3.5 cm. Substituting the values \(h = 16~cm\) and \(r = 3.5~cm\) into the formula for the volume of the cylinder gives:
Volume = \(\pi \times 3.5^2 \times 16 = 615.8~\text{cm}^3\) (to 1 decimal place)
Surface area
A cylinder has two circular faces at the ends and a curved surface.
The curved surface is like the label around a tin of soup – it can be flattened out to make a rectangle.
The length of the rectangle is the same as the circumference of the circular ends (\(2\pi r\)). The width of the rectangle is the same as the length of the cylinder (h).
The area of each of the two circles is \(\pi r^2\) and the area of the rectangle is \(2 \pi r \times h\).
The surface area of a cylinder is, therefore, \(\pi r^2 + \pi r^2 + 2\pi r \times h\).
This is more commonly written as:
\(\text{Total surface area of a cylinder} = 2 \pi r^2 + 2 \pi r h\)
Question
Calculate the total surface area of the cylinder.
Total surface area of a cylinder = the area of the two circular ends + the area of the curved surface.
Remember the curved surface unrolls like the label on a tin to make a rectangle with length equal to the circumference of the circle and width equal to the height of the cylinder.
The area of each circular end = \(\pi r^2 = \pi \times 3^2\).
The area of the curved surface = \(2\pi r \times h = 2 \times \pi \times 3 \times 10.\)
The total surface area = \(\pi \times 3^2 + \pi \times 3^2 + 2 \times \pi \times 3 \times 10 = 78\pi~cm^2 \) (in exact form) or 245 cm2 (to 3 sig.fig).
Notice that the calculation wasn't done until the end. If you work out answers along the way, such as the area of the circle, make sure you write down sufficient accuracy in your answers (usually at least four figures).