How to Calculate the Volumes of Basic Shapes

Let’s take a few minutes to review the volumes of shapes.

These get quite important in calculus problems because we often use calculus to do things like optimize areas, optimize volumes - you know, how much shampoo should you put in a shampoo bottle to make the most money?

that kind of thing.

I also like to call this the square peg in square hole problem.

And this is because of how I classify shapes.

Let’s start with 2D shapes.

So, now instead of looking at volume, let’s look at the area .

So, if you have a square with a side length s , then the area of the square is s ^2.

The perimeter , that is the length that borders the shape, is 4 s .

Now, a square is just a fancy type of rectangle.

A rectangle has some height, h , and some width, w , so the area of a rectangle is wh - the width times the height.

You can find the length of this line that surrounds the rectangle, or the perimeter, as 2 w + 2 h because we’re adding up w + h + w + h when we go around the rectangle.

Now, the second of our square peg in square hole area is the triangle.

So, let’s take a look at two different types of triangles.

We’ve got this guy here, and we’ve got this guy here.

Now both of these have a base that’s the width of the bottom part of the triangle, and it’s always going to be along some side.

We’re going to call that b .

Now the triangles also have some height.

It’s really important that the height be measured as being perpendicular to the base.

So, for some triangles, such as this one here, we need to measure the height actually outside of the base because we need this height to be straight up from the base, to be perpendicular from the base.

In these cases, the area is just 1/2 bh .

Now, I know this is all a review, so I won’t explain why.

Now, the last of my three types of shapes in my square peg, square hole is the circle.

We usually classify a circle by some radius , r , measured from the center of the circle straight out to the edge of the circle.

So, this radius is the same no matter where along the circle you measure it.

The area of a circle like this is pi r ^2.

Okay.

Square peg - square hole.

Rectangular peg - rectangular hole.

Triangular peg - triangular hole.

Circular peg - circular hole.

Well, that’s fine but these are all 3D shapes.

So this brings us to my two ways to classify 3D shapes.

The first, shapes that are the same from the top to the bottom.

So, here we go with the pegs.

We’ve got a can.

So, a can will fit in this circular hole and it will fit the same, no matter if it is the beginning of the can or the end.

There’s no difference.

It’s just a cylinder.

A cube is another example.

So, here the bottom of the cube - this cross-section here - looks just the same as the top.

Prism - same thing, except now your cross-section looks like a triangle.

The second classification for shapes, are shapes that aren’t the same from the top to the bottom.

So, these are things like a square pyramid, where we’ve got a square on the bottom and a point at the top, or a regular pyramid with a triangle on the bottom and a point on the top.

If I put the square pyramid into my hole, it will be a square at the bottom, but at the top it will just be a little point going through that hole.

Same thing with the triangular pyramid.

A sphere is another example.

If you slice through a sphere at the top, you’re going get a little, teeny, tiny circle.

At the very top you’re going to get no circle whatsoever.

It’s just going to be a point.

In the middle, the circle’s going to be big.

So imagine you’re taking an onion, and if you cut straight down, perpendicular to the cutting board, you’re going to end up with a lot of differently sized circles.

So, what does this mean in terms of finding the volume?

Well, for anything that has the same shape on the top and the bottom and all the way through, like the can, the cube and the prism, the volume is just the height times the area of the base.

So, here we’ve got our can.

We’ve got height.

The base area is just pi r ^2.

So, there’s my base.

So, the volume is h ( pi r ^2).

Say now I’ve got this box, this trunk of sorts, with a base that is a square.

So, now my base area is s ^2 because that’s the area of a square, and my volume is going to be the base area, s ^2 times h .

So my volume is h ( s ^2).

So, you can do this for anything where the cross-section is the same from the top to the bottom.

Say your base area is an E.

It looks like a big E.

Same thing.

Base area times the height will give you the volume.

So, what about things that change?

Let’s look at some of those.

For example, a sphere: the volume of a sphere is 4/3 pi r ^3.

A hemisphere is just half a sphere, and so the volume of a hemisphere is 1/2 (4/3 pi r ^3), which is also 2/3 pi r ^3.

Pyramids and cones are treated very similarly to one another.

So, each one has a height, and the volume is the h/3 (area of the base).

You can remember this for both of those.

So, the area of the base of a cone, obviously, is the area of this circle, and for a pyramid it’s going to be, in this case, a triangle, sometimes a square, but that’s going to be the area of the base.

So, let’s recap.

When you’re trying to find the volume of something just remember if it’s a square peg in a square hole.

If the base cross-section is the same as the cross-section throughout the entire volume then your volume is the base area, or one of those cross-sections, times the height.

If it’s not the same, then you need to remember one of the other rules.

For example, for cones, your volume is h/3 (base area).

For circles, just remember the equation for the volume of a sphere: 4/3 pi r ^3.