Difference between revisions of "Volume"
(expand) |
(→Cylinder: rmv extra $) |
||
| Line 12: | Line 12: | ||
The volume of a cylinder is <math>\pir^2h</math>, where <math>h</math> is the height<math> and </math>r<math> is the radius of the base. | The volume of a cylinder is <math>\pir^2h</math>, where <math>h</math> is the height<math> and </math>r<math> is the radius of the base. | ||
===Cone=== | ===Cone=== | ||
| − | The volume of a cone is </math>\frac 13\pir^2h<math>, where </math>h<math> is the height</ | + | The volume of a cone is </math>\frac 13\pir^2h<math>, where </math>h<math> is the height and </math>r$ is the radius of the base. |
== Problems == | == Problems == | ||
Revision as of 22:47, 11 December 2007
The volume of an object is a measure of the amount of space that it occupies. Note that volume only applies to three-dimensional figures.
Contents
Finding Volume
This section covers the methods to find volumes of common Euclidean objects.
Prism
The volume of a prism is 
, where 
is the area of the base and 
is the height.
Pyramid
The volume of a pyramid is given by the formula 
, where 
is the area of the base and is the height.
Sphere
The volume of a sphere is 
, where 
is the radius of the sphere at its widest point.
Cylinder
The volume of a cylinder is $\pir^2h$ (Error compiling LaTeX. Unknown error_msg), where is the height
r
\frac 13\pir^2h
h
r$ is the radius of the base.
Problems
Introductory
- Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron? (Source)
Intermediate
- A tripod has three legs each of length
feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 
feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let 
be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then can be written in the form 
where 
and 
are positive integers and is not divisible by the square of any prime. Find 
(The notation 
denotes the greatest integer that is less than or equal to 
) (Source)
feet. When the tripod is set up, the 
feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let 
where 
and 
are positive integers and 
(The notation 
denotes the greatest integer that is less than or equal to 
) (
