How to Find Surface Area, and Volume of a Right Cone.

Do you know that the volume of a right cone is one third the volume of a cylinder of the same height, and same base radius? Have you tried to find the volume, or the surface area of a cone, and gotten stuck when you realize that you just have the circumference of the base, and not the radius? Do you know the difference between the height of the cone, and its slant height? Do you know how to find the slant height if they just give you the circumference in the base, and the height of the cone?

Cones are the solids that require that you remember the Pythagorean Theorem, and/or special right triangles, or trigonometry. As the lesson advances, you will have the opportunity to practice the different cases that you may face when finding the surface area, and volume of cones. A special emphasis has been done in presenting every single stage in the solution of the examples. You will have the ability to test your understanding by solving the suggested problems following the examples. Enjoy a la carte!

Lesson's Content

Lesson's Glossary

Base of a conic solid
The planar region that forms the widest point of a conic solid; often labeled as the 'bottom' of the conic solid, it determines the exact shape of the conic solid.

Cone
It is a 3-D figure with circular base, a vertex not in the plane of the circle, and has a curved surface connecting the base with the vertex. The altitude of the cone is the perpendicular segment that goes from the vertex to the plane containing the circle at the base. The height is the length of the altitude. The slant height is the length of the distance from the vertex to the edge of the base. For a right cone it is necessary that the altitude contains the center of the circle at the base.

Conic solid
The set of points between a point (the vertex) and a non-coplanar region (the base), including the point and the region.

Diameter of a circle (or sphere)
The segment whose endpoints are points on a circle (or sphere) that contains the center of the circle as its midpoint; the length of that segment.

L.A.
Lateral area.

Lateral area
The area of the lateral surface of a solid.
 

Lateral surface
The surface not included in the base (s).   

Right cone
A cone whose axis is perpendicular to the plane containing its base.

Slant height
The length of a lateral edge of a conic solid

Solid
The union of the surface and the region of space enclosed by a 3-D figure; examples: conic solid, cylindrical solid, rectangular solid.

Solid geometry
The study of figures in three-dimensional space.

Space
The set of all possible points; made up of infinite planes.

 Surface
The boundary of a 3-D figure.

Surface area
The total area of the surface of a solid.

Unit cube
Unit of measuring volume.

Interactive Geometric Applets: Relevant Theorems.

Finding the surface area of cones requires that you know the slant

height, which is the shortest distance from a point in the circumference

in the base to the vertex in the cone. This slant height forms a right

triangle with the radius of the base, and the height of the cone.

Since you have a right triangle, then you may use the Pythagorean

Theorem. This applet presents you the process of finding lateral area,

surface area, and volume for the cone. You may manipulate "r", and "h"

to view different sets of problems.

A cone with a given height, and a given radius has a third of the volume of a cylinder

with that same height, and radius. Play with this applet to verify this volumen relationship.

The formulas are the same, with the only difference that we divide by three the formula

of the cylinder; to get the volume of the cone.

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