Work out the surface area of the prism.
The diagram shows a prism whose cross section is a trapezium or trapezoid. In order to find the surface area of this prism, we need to find the areas of each of its faces and then add them together. This prism has six faces of various different shapes, so we’ll calculate each of their areas.
Let’s begin with the only two faces that are the same, the front and the back of the prism. These two faces are each trapeziums. We can see this because the blue arrows indicate that two of the sides are parallel. To find the area of a trapezium, we find the average of the two parallel sides and then multiply by the perpendicular distance between them. This gives a calculation of six plus nine divided by two and then multiplied by four. Remember, there are two of these faces, the front and the back of the prism. So we’ll also multiply by two in order to find both areas. This gives a contribution of 60 to the total surface area from the front and the back.
Next, let’s think about the top of this prism. The top of the prism is in fact a square with sides of length six units. Therefore, its area is calculated by multiplying six by six. This gives a contribution of 36 to the total area from the top of the prism.
Next, let’s consider the area of the sloping face of the prism. This sloping face is a rectangle with sides of length six and five units. Therefore, its area is found by multiplying six by five, and so we have a contribution of 30 from the sloping face of the prism.
Now, there are two more faces of this prism that aren’t actually visible on the diagram. The first of these is the base. The base of the prism is another rectangle, this time with sides of length six and nine units. Therefore, its area is found by multiplying six by nine giving a contribution of 54 to the total surface area.
The final of the prism’s six faces is the vertical face. This is also a rectangle and its dimensions are six units and four units. It is this measurement of six units here that we’re using. So to find the area, we multiply six by four giving a contribution of 24 to the total surface area.
So we have the areas of each of the individual faces worked out. And to find the total surface area, we now just need to sum all of these areas together. So we have 60 plus 36 plus 30 plus 54 plus 24. Remember that 60 represents two of the faces of the prism. And we have a total of 204 for the surface area. Now we haven’t been given any units in the diagram, so this is just general area units.