EM - Volume of Prisms and Pyramids Lesson

Now let’s explore the volume of two other types of solids: pyramids and prisms. Both of these shapes are polyhedrons, which are three-dimensional figures whose faces are made of polygons.

Volume of Prisms

A prism is a polyhedron with two congruent faces, called bases, that lie on parallel planes. The other lateral faces that connect the bases are parallelograms.

The volume of a prism is the base area (B) times the height (h). $V=Bh$

Prisms are named by the shape of their bases. Here are some examples of different prisms with their bases shaded.

Triangular Prism		$V=Bh$ (Where h is the height of the prism.) (Area of triangular base) = $B=\frac{1}{2}bh$ (where h is the height of the triangle)
Rectangular Prism		$V=Bh$ (Where h is the height of the prism.) (Area of rectangular base) = $B=bh$ (where h is the height of the rectangle)
Square Prism		$V=Bh$ (Where h is the height of the prism.) (Area of square base) = $B=s^2$
Pentagonal Prism		$V=Bh$ (Where h is the height of the prism.) (Area of regular pentagonal base) = $B=\frac{1}{2}a\cdot5s$ (Where a is the apothem - distance from center to the side of the pentagon, and s is the side length.)
Octagonal Prism		$V=Bh$ (Where h is the height of the prism.) (Area if regular hexagonal base) = $B=\frac{1}{2}Pa$ (W)here P is the perimeter of the hexagon and a is the apothem (distance from the center of the hexagon to the side.)

The cross section that is parallel to the bases are congruent to the bases.

Prisms can be classified as right or oblique.

Right prisms have lateral sides that are perpendicular to the bases. Oblique prisms look slanted and the lateral sides are not perpendicular to the bases. In both types of prisms, the cross-sections parallel to the bases will always be congruent to the bases.

Right Prism	Oblique Prism

Volumes of Prisms Practice

Volume of Pyramids

A pyramid is a polyhedron with one base and lateral faces that are triangles. Pyramids are named by the shape of their base. Here are some examples.

Triangular Pyramid
Square Pyramid
Rectangular Pyramid
Pentagonal Pyramid

The volume of a pyramid is one-third the area of its base times the height. $V=\frac{1}{3}Bh$

The height (also called altitude) of the pyramid is measured from the vertex, perpendicular to the base. If the point of intersection is at the center of the base, it is called a right pyramid. If the intersection of the altitude and base is not at the center, it is called an oblique pyramid.

Right Pyramid	Oblique Pyramid