### Video Transcript

The solid shown is a rectangular prism with a triangular prism cut out along its length. Work out the volume of the solid.

So there’re probably two ways of approaching this problem, and we’ll discuss both. The first method is to subtract the volume of the triangular prism from the volume of the rectangular prism if it didn’t have this cutout through it.

In general, to find the volume of a prism, we find the area of its constant cross section first and then multiply this by the depth of the prism. The rectangular prism has a cross section of a rectangle, with dimensions of five and six, so its area was found by multiplying five by six. We then multiplied by the depth of the prism, which is seven.

The cross section of the triangular prism is a right-angled triangle, so its area was found by multiplying four by three and then dividing by two. To find the volume, we then multiply by the depth of the prism, which is again seven. The two volumes are 210 and 42 cubic units. And therefore, subtracting the volume of the triangular prism from the volume of the rectangular prism gives the volume of 168 cubic units, so that’s one method of answering this problem.

The other method is quite similar, but instead of subtracting the volume of the triangular prism from the volume of the rectangular prism, we instead subtract the areas and then multiply by the depth. What we’re doing is finding the area of the cross section of the prism of the cutout and then multiplying it by its depth.

So the area of the cross section would be six times five for the full rectangle, and then subtract four times three over two for the triangle that’s been cut out. We then multiply the area of the cross section by the depth of the prism, seven, in order to find the volume.

You can confirm that this does indeed give the same answer of 168 cubic units. So either of these two methods is absolutely fine; it’s just down to which you prefer to use to answer this question.