# Lesson Video: Volumes of Triangular Prisms Mathematics • 7th Grade

In this video, we will learn how to find volumes of triangular prisms and solve problems including real-life situations.

17:17

### Video Transcript

In this video, we’ll learn how to find volumes of triangular prisms and solve problems including real-life situations. We’ll consider what we actually mean by the word volume and how the properties of triangular prisms can help us to derive and use a formula for their volume. So, our first question is, what is a prism? A prism is a three-dimensional shape with a constant cross section. In other words, the cross section has the same shape and size throughout its length. A rectangular prism or a cuboid, for example, has a rectangular cross section. I could slice it down here or down here, and the size and shape of that rectangle would stay the same throughout its length. Similarly, a cylinder has a circular cross section.

Now, in this video, we’re actually interested in triangular prisms. These are, of course, prisms whose cross section is a triangle. And in this video, we’re learning how to calculate the volume of these shapes, where the volume is just the measure of its total three-dimensional space. Let’s consider how we find the volume of a cuboid or a rectangular prism and how that’s going to help us calculate the volume of a triangular prism. Imagine we have a rectangular prism. The dimensions of this prism are ℎ, 𝑤, and 𝑙 length units. We know that the formula that helps us calculate the volume of such prism is ℎ times 𝑤 times 𝑙. And that will be in cubic units.

Now, what we’re going to do is cut our prism perfectly in half from here to here. In doing so, we create two congruent, that means identical, triangular prisms. The volumes of each of these triangular prisms must be exactly half of the volume of the rectangular prism. We could write that as ℎ times 𝑤 times 𝑙 over two or, alternatively, a half times ℎ times 𝑤 times 𝑙. But let’s consider the first part of this formula, a half times ℎ times 𝑤. Do you recognize this formula? It looks a lot like the formula that helps us calculate the area of a triangle, and that’s because it is. And we can therefore rewrite the formula for the volume of a triangular prism. It’s the area of the triangular cross section multiplied by its perpendicular length.

Now, in fact, we can generalize this for any prism. Its volume is the area of its cross section times its perpendicular length. Now, whilst we can use either of these formulae, the second version is definitely the preferred form. Now, another way to think about this is that the volume or the number of cubic units that a slice with a length of one unit will hold will simply be the number of cubic units in that slice. Well, that’s just the same as the area of the cross section. Then, the length tells us how many of these slices we have, hence, area of the cross section times its length. Let’s begin with a simple example.

Determine the volume of the given triangular prism.

So, we’ve been given a triangular prism with a number of dimensions. Remember, a prism is just a three-dimensional shape with a constant cross section. And we know the formula that helps us calculate the volume of a triangular prism is the area of its triangular cross section multiplied by its length. Our cross section is in the shape of a triangle, so we also recall the formula for the area of a triangle. It’s a half times base times height. The base of our triangle measures 10 feet, whilst its height is six feet. So, its area is a half times 10 times six, which is 30.

Now, we’re working in feet, so the area measurement will be in square feet. Let’s now substitute everything we know into the formula for the volume of the prism. The area of its cross section is 30 while its perpendicular length, that’s the dimension that’s perpendicular to the cross section, is 17. So, the volume is 30 multiplied by 17, which is 510. Once again, since we’re working in feet, the volume will be cubic feet. And so, the volume of the given triangular prism is 510 cubic feet.

We’ll now look at how to find the volume of a triangular prism given information about the area of its cross section.

A prism has a triangular cross section whose area is 1.8 square meters. The length of the prism is 250 centimeters. Calculate the volume of the prism, giving your answer in cubic meters.

We’re given some information about a prism with a triangular cross section, in other words, a triangular prism. And we’re looking to calculate the volume of this prism. So, let’s recall the formula we need. The volume of a triangular prism is found by multiplying the area of its cross section by its perpendicular length. Well, we see that the area of our triangular cross section is 1.8 square meters. And the perpendicular length, remember, that’s the dimension that’s perpendicular to the cross section, is 250 centimeters.

Notice that we have two different units of measurement. We have square meters and centimeters. And so, before we calculate the volume, we either need to convert square meters into square centimeters or centimeters into meters. The question is actually telling us to give our answer in cubic meters, though. So, we’re going to convert the length measurement into meters.

We know that there are 100 centimeters in a meter, so to convert 250 centimeters into meters, we’ll divide by 100. 250 divided by 100 is 2.5. So, the length of our prism is 2.5 meters. Note that, had we instead wanted to convert our area into square centimeters, we would not have multiplied by 100 but by 100 squared. And so, we can see that whilst it’s possible, it is actually much easier to convert the length measurements rather than area or volume measurements.

We now substitute what we know about our prism into the formula for its volume. Its area is 1.8, and its length is 2.5, so the volume is the product of these. But how do we calculate 1.8 times 2.5? We could, of course, use a calculator, but let’s consider alternative methods. We’ll begin by calculating 18 times 25. There are a number of formal written methods to do so, but let’s use the column method. We begin by multiplying the eight by the five then the two, giving us 200 in the first row. We then multiply the one by the five and the two, remembering of course we’re really multiplying these by 10. So, we add a zero here, giving us 250 in our second row. Finally, we add these two numbers, giving us that 18 times 25 is 450.

So, how does this help us calculate 1.8 times 2.5? Well, 18 is 10 times larger than 1.8, as is 25 to 2.5. This means our answer, 450, is going to be 10 times and then another 10 times bigger or 100 times bigger. So, to find the value of 1.8 times 2.5, we’re going to divide by 100. 450 divided by 100 is 4.5. And remember, our answer is in cubic meters. And so, the volume of our triangular prism is 4.5 cubic meters.

In our next example, we’ll look at how we can find the unknown length of a triangular prism when given information about its volume.

The diagram shows a triangular prism whose volume is 1,932 cubic centimeters. Calculate the value of 𝑥.

We’ve been given some information about the volume of a triangular prism, so let’s begin by recalling the formula for this. The volume of a triangular prism is the area of the triangular cross section multiplied by its perpendicular length. We’re told that the volume is 1,932 cubic centimeters. We also know that the length that’s perpendicular to the cross section is 14 centimeters. So, we can therefore say that 1,932 must be equal to the area of the triangle multiplied by 14. Let’s solve this equation to find the area of the triangle in square centimeters.

We’ll divide through by 14. That tells us that the area of the triangle is 1,932 divided by 14 which is 138 or 138 square centimeters. Let’s now clear some space and remind ourselves on the formula for the area of a triangle. The area of a triangle is a half multiplied by the length of its base multiplied by its height. Now, it doesn’t actually matter that our triangle is scalene. We use the same formula. Remember, we calculated the area of our triangular cross section to be 138 square centimeters. The base is 𝑥 centimeters, and the height is 23 centimeters. So, we have another equation. 138 is a half times 𝑥 times 23.

This time, we’ll begin to solve this equation by multiplying both sides by two. 138 times two is 276. So, our equation becomes 276 equals 𝑥 times 23 or 23𝑥. Finally, we’re going to divide both sides of this equation by 23. So, 𝑥 is 276 divided by 23, which is 12. And so, given the information about our triangular prism and its volume, we find that 𝑥 in our diagram is equal to 12.

Which of the following has the largest volume?

We’ve been given four prisms. Remember, a prism is a shape which has a constant cross section. In order to compare their volumes, we’re going to begin by calculating the volume of each. And so, we recall that to find the volume of a rectangular prism and, in fact, a cube, we simply multiply its three dimensions. Our first shape is a cube, so its three dimensions are seven inches, seven inches, and seven inches. Its volume, therefore, must be seven times seven times seven or seven cubed. Seven cubed is 343. And since we’re working in inches, the volume of our first shape is 343 cubic inches.

Similarly, the volume of this rectangular prism is eight times six times five. We remember that multiplication is commutative. It can be performed in any order. Six times five is 30, so we need to work out eight times 30. And since eight times three is 24, we see that eight times 30 is 240. And the volume of our second shape is 240 cubic inches. But what about the triangular prisms? Well, the formula we tend to use is that it’s the area of the triangular cross section times its length, where the length is the length of the dimension perpendicular to the cross section. We also know that the area of a triangle is found by multiplying its base by its height and then dividing by two or a half base times height.

In our first prism, the one whose cross section is a scalene triangle, the length of its base is 12 inches and its height is nine inches. And so, the area of this triangle is a half times 12 times nine, which is 54 or 54 square inches. The length of the dimension perpendicular to this cross section is eight inches, so the volume of the shape is 54 times eight, which is 432 cubic inches.

Let’s repeat this process for our fourth prism. This time its cross section is a right-angle triangle, and the length of its base is 18, and its height is four. So, the area of this triangle is a half times 18 times four, which is 36 or 36 square inches. The length of the dimension perpendicular to this face is 10, so the volume of the shape is 36 times 10, which is 360. So, our final shape has a volume of 360 cubic inches. And we can say then that the shape which has the largest volume is the triangular prism whose cross section is a scalene triangle.

In our final example, we’ll look at some real-life applications of the volume of a triangular prism.

A vase in the shape of a triangular prism is standing on its triangular face as shown in the diagram. 200 cubic centimeters of water is poured into the vase. A bunch of flowers is then placed into the vase, causing the height of the water to rise by 1.8 centimeters. How much more water can be added to the vase until it is completely full? Give your answer in cubic centimeters.

So firstly, we note we are dealing with a triangular prism. It’s a prism whose cross section is in the shape of a triangle. We’re told that 200 cubic centimeters of water is initially poured into the vase, but that a bunch of flowers is placed in, which causes the height of the water to rise by 1.8 centimeters. We want to work out how much more space is in the vase essentially. So, we’re going to need to perform this in a few steps. Let’s begin by working out the height that the water reaches when 200 centimeters cubed is poured into the vase.

We begin by recalling that the volume of a triangular prism is found by multiplying the area of its triangular cross section by its length or its height. 200 cubic centimeters of water is poured into the vase, so the volume of our water is 200. We know the area of a triangle is a half multiplied by its base multiplied by its height. Well, the base of our triangle is five, and its height is eight, so its area is a half times five times eight, which is 20 square centimeters. We’re trying to work out the height, let’s call that ℎ, that the water reaches in the vase. So, we have an equation: 200 equals 20 times ℎ.

Let’s find the value of ℎ by dividing both sides of this equation by 20. ℎ is therefore 200 divided by 20, which is 10 or 10 centimeters. We can say then that the initial height of water in the vase is 10 centimeters. We’re now going to work out the height of water in the vase when the flowers are added. So, that must be 10 plus 1.8, since it rises by 1.8. And we can now say that ℎ sub two, the second height of water, is 11.8 centimeters. We essentially want to work out the remaining volume in the vase. So, we’re going to find the difference between the total height of our vase and the height that the water now reaches. That’s 14 minus 11.8, which is 2.2 centimeters.

So, we need to work out the volume of the slice of the vase, which is 2.2 centimeters high. Remember, we already calculated the area of the triangular cross section to be 20. And the whole point of a prism is that its cross section doesn’t change. So, the area of the cross section we’re interested in is also 20. And the volume is 20 times 2.2, which is 44 or 44 cubic centimeters. And so, 44 cubic centimeters of water can be added to the vase to make it completely full.

Now, interestingly, this wasn’t the only way we could’ve answered this problem. Instead, we could’ve worked out the volume of this slice. And in fact, had we done so, we would’ve found that the volume of this slice is 36 cubic centimeters, meaning that after the flowers are added, we have a total of 200 plus 36. That’s 236 cubic centimeters of volume being used. We would have then had to have calculated the volume of the entire vase. Had we done that, we would have found it to be 280 or 280 cubic centimeters. The amount of space left in the vase is 280 minus 236 which is, once again, 44 cubic centimeters.

In this video, we’ve learned that a prism is a three-dimensional shape with a constant cross section. We saw that we can find the volume of a triangular prism by multiplying the area of its triangular cross section by its length and that we need to be careful to ensure that our units match before we perform any calculations.

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