A prism is a three-dimensional solid figure with two identical ends. It is made up of flat sides, similar bases, and equal cross-sections. Its faces are parallelograms or rectangles without bases. Such a prism that has three rectangular faces and two parallel triangle bases is called a triangular prism. The triangular bases are connected by lateral faces that run parallel to one another.
Volume of a Triangular Prism Formula
A triangular prism’s volume is defined as the space inside it or the space filled by it. Knowing the base area and height of a triangular prism is all that is required to calculate its volume. The volume of a triangular prism is equal to the product of the base’s area and the prism’s height, also known as the length of the prism. The base area of a triangular prism is equal to half of the product of the triangular base and its altitude.
Formula
V = (1/2) × b × h × l
where,
b is the triangular base,
h is the altitude of the prism,
l is the length of prism.
Sample Problems
Problem 1. Find the volume of a triangular prism if its base is 6 cm, altitude is 8 cm and length is 12 cm.
Solution:
We have, b = 6, h = 8 and l = 12.
Using the formula we have,
V = (1/2) × b × h × l
= (1/2) × 6 × 8 × 12
= 3 × 8 × 12
= 288 cu. cm
Problem 2. Find the volume of a triangular prism if its base is 5 cm, altitude is 7 cm and length is 8 cm.
Solution:
We have, b = 5, h = 7 and l = 8.
Using the formula we have,
V = (1/2) × b × h × l
= (1/2) × 5 × 7 × 8
= 5 × 7 × 4
= 140 cu. cm
Problem 3. Find the length of the triangular prism if its base is 6 cm, altitude is 9 cm and volume is 98 cu. cm.
Solution:
We have, b = 6, h = 9 and V = 98.
Using the formula we have,
V = (1/2) × b × h × l
=> 98 = (1/2) × 6 × 9 × l
=> 196 = 27l
=> l = 196/27
=> l = 7.25 cm
Problem 4. Find the altitude of the triangular prism if its base is 8 cm, length is 14 cm and volume is 504 cu. cm.
Solution:
We have, b = 8, l = 14 and V = 504.
Using the formula we have,
V = (1/2) × b × h × l
=> 504 = (1/2) × 8 × h × 14
=> 504 = 56h
=> h = 504/56
=> h = 9 cm
Problem 5. Find the area of the base of the triangular prism if its length is 18 cm, height is 10 cm and volume is 450 cu. cm.
Solution:
We have, l = 18, h = 10 and V = 450.
Using the formula for volume we have,
V = (1/2) × b × h × l
=> 450 = (1/2) × b × 10 × 18
=> 450 = 90b
=> b = 450/90
=> b = 5 cm
Hence, the area of triangular base is,
A = (1/2) × b × h
= (1/2) × 5 × 10
= 25 sq. cm
Video Transcript
Determine the volume of the given triangular prism.
The volume of a triangular prism can be found by multiplying the base times the height, where the shaded pink portion represents the base. The volume is then the area of the base multiplied by the height. And the green portion represents the height. It’s the distance from one base to the other.
And how do we go about finding the area of the base? Well, like any triangle, we multiply one-half, the base of that triangle, times the height of that triangle. So we see, in our case, the base of our triangle is 10 feet and the height of our triangle is six feet. And the height of our triangular prism is 17 feet.
Let’s start plugging things into our formula. For our base, our triangle, one-half times six times 10. Then we multiply the area of our base by the height of our prism, 17 feet. Now we simply multiply. six times 10 times one-half equals 30. And 30 times 17 equals 510.
But we’re not finished here because we need to decide what to do with our units. When we multiply feet by feet by feet and when we’re discussing volume, we know that our units will be cubed. That’s our final answer: The volume of the given triangular prism equals 510 feet cubed.
Video transcript
Let's do some solid geometry volume problems. So they tell us, shown is a triangular prism. And so there's a couple of types of three-dimensional figures that deal with triangles. This is what a triangular prism looks like, where it has a triangle on one, two faces, and they're kind of separated. They kind of have rectangles in between. The other types of triangular three-dimensional figures is you might see pyramids. This would be a rectangular pyramid, because it has a rectangular-- or it has a square base, just like that. You could also have a triangular pyramid, which it's just literally every side is a triangle. So stuff like that. But this over here is a triangular prism. I don't want to get too much into the shape classification. If the base of the triangle b is equal to 7, the height of the triangle h is equal to 3, and the length of the prism l is equal to 4, what is the total volume of the prism? So they're saying that the base is equal to 7. So this base, this right over here is equal to 7. The height of the triangle is equal to 3. So this right over here, this distance right over here, h, is equal to 3. And the length of the prism is equal to 4. So I'm assuming it's this dimension over here is equal to 4. So length is equal to 4. So in this situation, what you really just have to do is figure out the area of this triangle right over here. We could figure out the area of this triangle and then multiply it by how much you go deep, so multiply it by this length. So the volume is going to be the area of this triangle-- let me do it in pink-- the area of this triangle. We know that the area of a triangle is 1/2 times the base times the height. So this area right over here is going to be 1/2 times the base times the height. And then we're going to multiply it by our depth of this triangular prism. So we have a depth of 4. So then we're going to multiply that times the 4, times this depth. And we get-- let's see, 1/2 times 4 is 2. So these guys cancel out. You'll just have a 2. And then 2 times 3 is 6. 6 times 7 is 42. And it would be in some type of cubic units. So if these were in-- I don't know-- centimeters, it would be centimeters cubed. But they're not making us focus on the units in this problem. Let's do another one. Shown is a cube. If each side is of equal length x equals 3, what is the total volume of the cube? So each side is equal length x, which happens to equal 3. So this side is 3. This side over here, x is equal to 3. Every side, x is equal to 3. So it's actually the same exercise as the triangular prism. It's actually a little bit easier when you're dealing with the cube, where you really just want to find the area of this surface right over here. Now, this is pretty straightforward. This is just a square, or it would be the base times the height. Or essentially the same, it's just 3 times 3. So the volume is going to be the area of this surface, 3 times 3, times the depth. And so we go 3 deep, so times 3. And so we get 3 times 3 times 3, which is 27. Or you might recognize this from exponents. This is the same thing as 3 to the third power. And that's why sometimes, if you have something to the third power, they'll say you cubed it. Because, literally, to find the volume of a cube, you take the length of one side, and you multiply that number by itself three times, one for each dimension-- one for the length, the width, and-- or I guess the height, the length, and the depth, depending on how you want to define them. So it's literally just 3 times 3 times 3.