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## High school geometry

### Course: High school geometry > Unit 9

Lesson 3: Volume and surface area# Volume of triangular prism & cube

Using the formulas for the volume of triangular prism and cube to solve some solid geometry problems. Created by Sal Khan.

## Want to join the conversation?

- So the formula for the volume of solid geometric shapes is the area of one side times the depth/length?(276 votes)
- For prisms and cylinders... yes, Area of Base times Height. For Pyramids and Cones its 1/3 of that.(245 votes)

- What is the formula for the volume of a rectangular pyramid?(12 votes)
- Volume of all types of pyramids = ⅓ Ah, where h is the height and A is the area of the base. This holds for triangular pyramids, rectangular pyramids, pentagonal pyramids, and all other kinds of pyramids.

So, for a rectangular pyramid of length ℓ and width w:

V = ⅓ hwℓ (because the area of the base = wℓ )(19 votes)

- Can you use algebra to find any geometric answers? Ex. Can you use algebra to find the volume of a cube/cone?(6 votes)
- Well, you can't really use algebra to find the volume of anything. The only basic thing to do to find volume is to use the formula for the given shape. What you can use algebra for, is when you want to find the height, radius, or length of a 3-Dimensional shape. For example, to find the radius of a cylinder, you need to know all the other dimensions, such as what pi is defined to (often 3.14) , the height of the shape, and the volume of the shape (the formula for a cylinder is pi times the radius squared times the height). You put the known dimensions into the formula, and then solve for the missing variable, the radius. So you can use algebra in some way in geometry, but only to find a missing height, radius, length, etc. of a 3-Dimensional shape.(9 votes)

- Shouldn't the answer to the 1st problem be 84?(2 votes)
- You've probably already figured it out by now, but for those that have the same question, here is the answer:

The equation for finding the volume of a triangular prism is:

½ × b × h × l = Volume

This means that the equation for the 1st problem would've been:

½ × 7 × 3 × 4 = 42

And you probably just forgot to multiply your equation by ½:

7 × 3 × 4 = 84

If we had a**rectangular prism**with the same length, width, and height, then yes, the answer would be 84 because we wouldn't have to multiply by ½.(3 votes)

- hello i am new to geometry i would like to know tips on it(4 votes)
- Khan is a life saver i got an a+ thanks!(4 votes)
- okay I have a question.......if I have a rectangular prism that has a volume of 18 cubic feet and a length of 6 feet and also the width of 2 feet ...........then what is the height?(2 votes)
- Volume is length times width times height. 18=12h so h is 1.5. There is no 1/2 involved : )(2 votes)

- what? i don't understand(2 votes)
- What don't you understand? Tell me and I will help you :)(5 votes)

- Wouldn't that mean that x to the 4th power would be x tesseracted?(5 votes)
- While x to the 4th power is not usually called x tesseracted, that would certainly be a good name for x to the 4th power.(1 vote)

- So the formula for the volume of solid geometric shapes is the area of one side times the depth/length?(2 votes)
- While your language is not quite correct, you have the right idea. First, a geometric solid can be many things including pyramids, spheres, etc. which have different rules for volume. However, if it is a prism (including a cylinder), it is the area of a base, which is one of two parallel and congruent shapes (not just any side), times the height.(4 votes)

## 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.