Volume Formulas for Pyramids, Prisms, Cones & Cylinders

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  • 0:03 Volumes of…
  • 1:08 Pyramid and Prisms
  • 3:44 Cones and Cylinders
  • 6:18 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Pyramids, prisms, cones, and cylinders are three-dimensional objects that show up everywhere in the world around us. This lesson teaches us how to find the volume of these objects and look at the relationships between their volumes.

Volumes of 3-Dimensional Objects

Suppose you're moving, and you are packing up your things in boxes. You know the dimensions of the boxes, but you want to know how much of your stuff you can fit in each box. That space inside the box is called the volume of the box. We're going to see how to find it for different objects using formulas and relationships.

Three-dimensional objects are objects that have a length, a width, and a height. We live in a three-dimensional world, so there are three-dimensional objects all around us. For example, the buildings we work in and the houses we live in are three-dimensional objects. As we said, the volume of a three-dimensional object is the amount of space that is inside the object.

We're going to look at four specific three-dimensional objects. Those are pyramids, prisms, cones, and cylinders. We will look at how to find the volume of each of these using a formula. In the process, we will see the relationship between the volumes of pyramids and prisms, along with the relationship between the volumes of cones and cylinders.

Pyramids and Prisms

A pyramid is a three-dimensional object with triangles as faces and a polygon base. The faces are sides, and the base is the bottom. A polygon is a two-dimensional shape with straight sides. A perfect real-world example of pyramids are the pyramids in Egypt.

A prism is a three-dimensional object with two polygon bases that are the same shape and size, and faces that are rectangles. An example of a prism in the real world is a cardboard box.

When we have a prism and a pyramid with equal bases and height, we can fit the pyramid inside the prism. As a matter of fact, when this is the case, the pyramid takes up exactly 1/3 of the space in the prism. This fact allows us to see a relationship between the volume of a prism and the volume of a pyramid when the two have equal bases and equal heights. That is, the volume of a pyramid is 1/3 the volume of a prism with equal height and base.

In finding the volume of a prism, we want to find how much space is on the inside of the prism. To do this, we can take the area of one of the bases and multiply it by the height. So:

  • Volume of a prism = (area of base) * height

Now, let's consider the volume of a pyramid. As we said, a pyramid takes up 1/3 of the volume of a prism when their bases and height are equal. Therefore, the volume of a pyramid is 1/3 multiplied by the volume of a prism. So:

  • Volume of a pyramid = 1/3 (area of the base) * height

Suppose we have a prism with a base area of 16 square inches. The height of the prism is 8 inches. We can use our formula to find the volume of the prism:


Now, consider a pyramid with a square base that has an area of 16 square inches, and height of 8 inches. We can find the volume of the pyramid using the formula, or we can just multiply the volume of the prism by 1/3.


In both cases, we find that the volume of the pyramid is 42.7in 3.

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