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Volume of Pyramids - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

The formula for the volume of prisms can also be applied to the volume of pyramids, which occupy only one third of the space of a corresponding prism with the same base and height. To calculate the volume of a pyramid, we simply multiply the area of the one of the bases times one third of the height of the pyramid. This formula is very similar to volume of cylinder and volume of cone formulas.

When we're talking about the volume of a pyramid. That is the space that's enclosed by the three dimensional figure of a pyramid, we can compare it to a rectangular prism that is the same height and the same base. But notice in a pyramid, we only have one polygon that's a base.
We said that the volume of a rectangular prism is going to equal the base area times the height. So I'm going to label the height, capital h as the height of your solid. The reason why I use the capital H here is because, if you had a trapezoid for a base, you're going to have a lower case h because that trapezoid will have its own height plus you'll have the height of your solid. So the relationship between the volume of a rectangular prism and a pyramid where they have the same height is that this volume will be one third the volume of this rectangular prism.
Very similar to the relationship of cones and cylinders. So if you have a pyramid or a cone, you're going to have one third base area times height. Now the difference between a cone and a pyramid, is that with the cone we could say this base area will always be pi r squared. However, in calculating the volume of the pyramid, you have to ask yourself what type of polygon is the base? So if you had a regular hexagon, for this base area you're going to say one third, apothem times side length times the number of sides divided by 2 times your height. So you have to be careful about what you use for your base area formula.

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