Geometry

Pyramids and Cones

Big Picture

Pyramids and cones have sides that join at a single point. The properties of pyramids and cones are often used in real life. With surprisingly simple formulas, we can find the volume and surface area of these objects.

Key Terms

Pyramid: A polyhedron with one base and triangular sides meeting at a common vertex.

Slant Height: The height of a lateral face of a regular pyramid.

Cone: A solid with a circular base and sides tapering up towards a common vertex.

Pyramids

A pyramid has one base and all the lateral faces meeting at a vertex.

When the vertex is directly above the center of the base and the base is a regular polygon, the pyramid is regular. A regular pyramid has a slant height, which is the height of the lateral face. Non-regular pyramids do not have a slant height.

Pyramids

Surface Area of a Regular Pyramid

Theorem: For a regular pyramid with a base of area B, perimeter P, and slant height l, then the surface area is:

  • \(SA = B + \frac{1}{2}PI\)
Surface Area of a Regular Pyramid

Volume of a Pyramid

Theorem: The volume of a pyramid is \(V = B + \frac{1}{3}Bh\), where B is the area of the base and h is the height.

The volume of a pyramid is one-third the volume of a prism with the same base.

Cones

A cone is a solid with a circular base and sides that taper up towards a common vertex. The radius of the base is also the radius of the cone.

If the segment connecting the vertex of the cone to the center of the base is perpendicular to the base, then the cone is a right cone. It has a slant height just like a pyramid; the slant height is the distance between the vertex and a point on the base.

cone
Image Credit: Blue figures on this page copyright rt guest, 2014, modified by CK-12 Foundation. Used under license from Shutterstock.com

Geometry

Pyramids and Cones Cont.

Cones (cont.)

Surface Area of a Right Cone

The surface area of a right cone can be tricky to calculate. The net of a cone looks like the diagram below:

Surface Area of a Right Cone

The area of the lateral face is a sector and can be found by using the following proportion:

\(\frac{\text{Area of circle}}{\text{Area of sector}} = \frac{\text{Circumference}}{\text{Arc length}}\)

\(\frac{\pi l^2}{\text{Area of sector}} = \frac{2 \pi l}{2 \pi r} = \frac{l}{r}\)

\(\text{Area of sector} = \pi r l\)

Theorem: The surface area of a right cone with base radius r and slant height h is \(SA = \pi r^2 + \pi rl\).

Volume of a Cone

Theorem: The volume of a cone is \(V = \frac{1}{3} \pi r^2 h\), where r is the radius of a cone and h is the height.

The volume of a cone is one-third the volume of a cylinder with the same base.

Notes