NSW Mathematics Standard 11 - 2020 Edition

5.01 Surface area of prisms and pyramids

Lesson

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 (mm^{2}), square centimetres (cm^{2}), square metres (m^{2}), and square kilometres (km^{2}).

Here, we look at how to calculate the surface area of two specific categories of 3D shapes: prisms and pyramids.

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.

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:

Surface area of a prism

$\text{Surface area of a prism }=\text{area of ends }+\text{area of connecting faces }$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.

Converting units of area

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 m^{2} |
$=$= | $100\times100$100×100 cm^{2} |

$=$= | $10000$10000 cm^{2} |

Consider the following cube with a side length equal to $6$6 cm. Find the total surface area.

Find the surface area of the triangular prism shown.

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 m^{2}.

A pyramid is a 3D shape that has a polygon as a base and sloping sides that meet at a point called the apex. If the apex is directly above (perpendicular to) the centre of the base, the pyramid is called a right pyramid. These following are all right pyramids. Pyramids are named by the shape of their base.

As we can see from these diagrams, the triangular shaped sides slope towards the apex. This introduces a new term we use in calculations with pyramids called the **slant height **(or slope height).

In the following applet, notice that the slant height corresponds to the height of the triangle that makes up each face. Sometimes we need to calculate the slant height using Pythagoras' theorem.

As we found with prisms, calculating the surface area of a solid is done by adding the area of all faces. For right pyramids, we have the base and a number of triangular faces.

This results in:

Surface Area of Right Pyramid

$\text{Surface area of right pyramid }=\text{area of base }+\text{area of triangles }$Surface area of right pyramid =area of base +area of triangles

Find the surface area of the square pyramid shown. Include all faces in your calculations.

Find the surface area of the rectangular pyramid shown. Include all faces in your calculations.

Consider the following square pyramid:

Find the length of the slant height.

Round your answer to two decimal places.

Using the value for slant height found in part (a), find the surface area of the square pyramid. Make sure to include all faces in your calculations.

Round your answer to one decimal place.

performs calculations in relation to two-dimensional and three-dimensional figures