Measurement

Ontario 10 Applied (MFM2P)

Surface area of prisms

Lesson

As the words suggest surface area, is the area of a surface. Surface area is very useful. Here are some examples:

In **construction**, surface area affects planning (how much to buy) and costs (how much to charge) in connection with such things as wallboard, shingles, and paint.

In **manufacturing **you will have the same issues - say, the cost of making boxes or printing, or sheet metal parts.

In **designing**, surface area enters into calculations of wind resistance and drag in cars or airplanes, pressure and strength of materials.

The surface area exposed to air affects how fast something cools or heats or dries out. Elephants, for example, need big ears to increase their surface area for cooling purposes, so it's not only human designers that have to do these calculations.

Many objects have complex shapes to increase their surface area: the inside of your lungs, intestines, and brain, air cleaners, radiators - and towels. Others avoid flat shapes to minimize surface area and avoid drying out: pine needles and cactus for example.

Many things you buy for home use are priced by surface area - or, if not, you should figure out their cost per square metre to decide which is the best buy.

We have already been introduced to the idea of solid objects such as prisms. We saw these when we were playing with volumes (here and here). So we can recognise these sorts of shapes as being prisms.

A prism has two end pieces which are congruent (exactly the same).

It also has a number of faces that join the 2 end pieces together.

For example, this triangular prism has 2 triangular end pieces and then 3 faces. We could see that this shape would have a net that looked like this.

The surface area of this shape will be the sum of the area of all the faces. This is the same as the total area of the net.

**2 triangle pieces and 3 rectangular pieces**

We know how to find the area of a triangle, so these pieces will be easy. The three rectangles have dimensions equal to the lengths of the sides of the triangle and width equal to the height of the prism.

Have a look at this interactive to see how to unfold prisms.

So when needing to calculate the surface area (SA) of a prism you need to add up the areas of individual faces. Take care not to miss faces or double up and look for clever methods too, like 2 faces that might have the same area!

Surface Area of a Prism

$\text{Surface Area of a Prism }=\text{Sum of areas of all faces}$Surface Area of a Prism =Sum of areas of all faces

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

Given the following triangular prism. Find the total surface area.

Find the surface area of this prism.

Solve problems involving the surface areas of prisms, pyramids, and cylinders, and the volumes of prisms, pyramids, cylinders, cones, and spheres, including problems involving combinations of these figures, using the metric system or the imperial system, as appropriate