Answer the following questions for the prism shown. Work out the volume of the prism. Work out the surface area of the prism.
Remember, a prism is a three-dimensional shape with a constant cross section. The cross section is the shape that’s made by cutting straight across an object. Now, in this example, if we cut in this direction, we end up with that constant cross section. Here, that’s an L shape. Now, that’s really important because the first part of this question asks us to find the volume of this prism. And so we recall the formula to find the volume of a prism is the area of its cross section multiplied by its length, or alternatively its height. And so we’re going to need to work out the area of this L shape. And then we’ll multiply it by the length or the height of the prism.
Now, let’s redraw the L shape, so we’re looking at it face on. This is called a composite or a compound shape. And we call it this because it’s made up of two or more polygons. In this case, we can split this shape up into two rectangles. There are a number of ways to do this, but let’s do it by dropping this vertical line in here. To find the total area of the composite shape then, we’re going to find the area of the two separate rectangles. Now, the area of a rectangle is simply its width multiplied by its height or the two sides multiplied together. So here, that’s two times six.
But what about the area of this second rectangle? We can see that one of its dimensions is three units. But what’s its other dimension? A common mistake here is to think that it’s exactly half the length of the side parallel to it. In fact, we need to take into account this other side. We can see the total width of the shape is five units and the width of the other rectangle is two units. This means the width of this rectangle must be the difference between these. It’s five minus two which is three.
And so the area of this rectangle is three multiplied by three. And the area of the cross section is therefore 12 plus nine which is 21 or 21 units. And the volume of this shape is therefore this area multiplied by its length. This is the length of the edge that’s perpendicular to that plane, to that cross section. And here, that’s four. So the volume is 21 times four which is 84 or 84 cubic units.
The second part of this question asks us to find the surface area of the prism. That’s the combined area of all of its faces. We’ve actually found the area of two of its faces already. There’s an L shape at the front and the back of our prism. But what about the other faces? We have a rectangle here and one up here that we can see. There’s another rectangle at the base of our shape, two at the back. And then rectangle eight is here; it’s almost on a step. The surface area is the combined area of all of these faces. So let’s calculate them.
We have the two L shapes, and we already calculated their area to be 21 units. We then have rectangle three, which is at the front of our shape. The area of a rectangle is the product of its two dimensions. So here, that’s four units and six units. Rectangle four is up here. One of its dimensions we see to be two units. But actually the depth or what we called the length of our prism remains constant. So its other dimension is four units. And the area of rectangle four is two times four.
Then there’s a rectangle at the very bottom of our shape, rectangle five. It has, once again, one measurement of four units, but its other dimension is five units. So its area is four times five. Rectangle six at the back here has a dimension of three units. Its other dimension is four units. Again, it’s the length of our prism. But what about the dimensions of rectangle seven? Well, one of its dimensions is four units. It’s the length of the prism. But if we go back to our sketch of the L shape, its other dimension is this length here. This is the difference between six and three units, so it’s three units. And so the area of rectangle seven is three times four.
Now, rectangle eight is the same. Remember, we calculated this measurement here to be three units. Its other dimension is, once again, given by the length of the prism. And so our surface area is found by adding together 21, 21, 6 times four, two times four, four times five, three times four, three times four, and another three times four. That gives us a total value of 130 or 130 square units.
And in fact, there was an alternative method we could’ve used here. We notice that each rectangle had one dimension that was four units. And so what we could have done is almost imagine that we were dismantling the net of this shape. When we do so, we end up with a really long rectangle. One of its dimensions is four units. Its other dimension is the total perimeter of our L shape. That’s two plus six plus five plus three plus three plus three, which is 22 units. And so we end up calculating the area of that rather long rectangle to be 22 times four. And then we add it to our two areas for the L shapes. Either way, we do indeed get 130 square units.