Surface area is the sum of the areas of all the faces of a three dimensional (3D) shape, and is measured in units such as square millimetres (mm2), square centimetres (cm2), square metres (m2), and square kilometres (km2).
Here, we look at how to calculate the surface area of prisms.
A useful strategy when calculating surface areas is, working methodically, to separate the 3D body into its 2D component faces. Often faces are repeated and we only need to work out the area of a repeated face once before multiplying it by the number of occurrences. We must take care when determining the dimensions of each of the component faces, and ensure that we don't miss any of the faces in our calculations.
Surface area of prisms
A prism is a 3D shape that has two identically shaped ends. It is named according to shape of the end faces.
As shown above, a prism has two end pieces which are exactly the same. It also has a number of flat faces that join the two ends together.
For example, the triangular prism below has two triangular ends and three rectangular faces. If we know how to calculate the area of these individual shapes then we can calculate the surface area. The net of this shape is also shown and can assist in the visualisation and calculation of the surface area.
The following applet shows how a rectangular prism unfolds into a net of the prism.
The formula for the surface area of a prism is:
$\text{Surface area of a prism }=\text{area of ends }+\text{area of connecting faces }$Surface area of a prism =area of ends +area of connecting faces
Three-dimensional shapes can be considered open or closed. For an open shape we must only include the actual faces required in our surface area calculations.
When converting between units of area, we need to multiply by the conversion factor for each dimension of the shape. That is, for 2D shapes we need to multiply by the conversion factor twice. For example:
| If $1$1 m | $=$= | $100$100 cm |
| then $1$1 m2 | $=$= | $100\times100$100×100 cm2 |
| $=$= | $10000$10000 cm2 |
Practice questions
QUESTION 1
Consider the following cube with a side length equal to $6$6 cm. Find the total surface area.
QUESTION 2
Find the surface area of the triangular prism shown.
QUESTION 3
A swimming pool has the shape of a trapezoidal prism $14$14 metres long and $6$6 metres wide. The depth of the water ranges from $1.2$1.2 metres to $2.5$2.5 metres, as shown in the figure.
Calculate the area inside the pool that is to be tiled (assuming that the top of the pool will not be tiled).
Give your answer to the nearest $0.1$0.1 m2.
