Video: Finding the Volume of a Composite Prism

The figure shows the design of a swimming pool. Work out, in cubic meters, the volume of water needed to fill the swimming pool completely.

03:06

Video Transcript

The figure shows the design of a swimming pool. Work out, in cubic meters, the volume of water needed to fill the swimming pool completely.

Now, this swimming pool is a composite solid. And we need to look carefully at the diagram to identify the individual solids it’s composed of. The shallow end of this swimming pool is a cuboid or rectangular prism with dimensions of 10 meters, 15 meters, and 1.5 meters. The deep end of the swimming pool is another cuboid with dimensions of 15 meters, four meters, and 15 meters. We know that for a cuboid, the volume is equal to the length multiplied by the width multiplied by the height. So, we can determine the volume of water needed to fill the deep end and the shallow end of the swimming pool. The volumes are 900 and 225 cubic meters, respectively.

Now, where it gets a little trickier is the sloping part in the middle of the pool. But if we look carefully, we can see that this sloping part is in fact a prism. The cross section of this prism is this face now highlighted in green, and it is a trapezium or trapezoid. We know that the volume of a prism is the area of its constant cross section multiplied by the depth or height of the prism. And we should also recall that the area of a trapezoid is half the sum of the parallel sides, which we often refer to as 𝑎 and 𝑏 multiplied by the perpendicular distance between them, which we often refer to as ℎ.

Let’s look carefully at the diagram to determine each of these values. The two parallel sides of our trapezoid are the depths of the deep and shallow ends of the pool. They are four meters and 1.5 meters. The perpendicular distance between these two sides can be found by subtracting the length of the shallow end and the length of the deep end, which are 10 meters and 15 meters, from the total length of the pool, which is 30 meters. That gives five meters. The depth of this prism, not to be confused with the depth of the pool, is the same as the depth of the other two. It’s 15 meters.

So the volume of this trapezoid or prism, it’s a half multiplied by four plus 1.5 multiplied by five, for the area of the cross section, and then multiplied by 15, which is the depth of the prism. When evaluated, that gives 206.25, and again the units are cubic meters. The total volume of water needed to completely fill the swimming pool then is found by adding these three values together. That’s 1,331.25 cubic meters.

A lot of the skill involved in this question was in looking carefully at the diagram to identify the individual solids that made up this composite solid and, of course, their individual dimensions. Using color, as I did in this example, can really help with this.

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