How to Calculate the Volumes of Basic Shapes

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Volume of a Frustum of Pyramids & Cones

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
Your next lesson will play in 10 seconds
  • 0:12 Square Peg in a Square Hole
  • 0:38 2D Shapes
  • 2:56 3D Shapes
  • 4:41 Volume of 3D Shapes
  • 6:34 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Speed Speed
Lesson Transcript
Instructor: Heather Higinbotham
Squares pegs = square holes. Triangular pegs = triangular holes. But where does a sphere go? In this lesson, review volumes of common shapes while contrasting a sphere and a cylinder - after all, they both go into the circular hole... right?

Square Peg in a Square Hole

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.

2D Shapes

Formula for finding the area of triangles
2D Triangle Area

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 s2. 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 r2.

3D Shapes

Spheres are 3D shapes that do not have the same dimensions from top to bottom
Sliced Sphere

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.

To unlock this lesson you must be a Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use

Become a member and start learning now.
Become a Member  Back
What teachers are saying about
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account