Volume of Right Prisms

Student Summary

Any cross-section of a prism that is parallel to the base will be identical to the base. This means we can slice prisms up to help find their volume. For example, if we have a rectangular prism that is 3 units tall and has a base that is 4 units by 5 units, we can think of this as 3 layers, where each layer has 454\boldcdot 5 cubic units. The volume of the figure is the number of cubic units that fill a three-dimensional region without any gaps or overlaps.

Two images. First, a prism made of cubes stacked 5 wide, 4 deep, 3 tall. Second, each of the layers of the prism is separated to show 3 prisms 5 wide, 4 deep, 1 tall.

That means the volume of the original rectangular prism is 3(45)3(4\boldcdot 5), or 60, cubic units.

This works with any prism! If we have a prism with a height of 3 cm that has a base with an area of 20 cm2, then the volume is 3203\boldcdot 20 cm3 regardless of the shape of the base. In general, the volume of a prism with height hh and area BB is

V=Bh\displaystyle V = B \boldcdot h

For example, these two prisms both have a volume of 100 cm3.

Prism with triangular base, area 20 centimeters squared, and height 5 centimeters.

Prism with irregular base, area 25 centimeters squared, and height 4 centimeters.

Visual / Anchor Chart

Standards

Addressing
7.G.6

Solve real-world and mathematical problems involving area of two-dimensional objects composed of triangles and trapezoids. Solve surface area problems involving right prisms and right pyramids composed of triangles and trapezoids. Find the volume of right triangular prisms, and solve volume problems involving three-dimensional objects composed of right rectangular prisms.