Difference between revisions of "Volume"
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The volume of a [[cone]] of height <math>h</math> and radius <math>r</math> is <math>\frac{\pi r^2h}{3}</math>. (Note that this is just a special case of the formula for a pyramid.) | The volume of a [[cone]] of height <math>h</math> and radius <math>r</math> is <math>\frac{\pi r^2h}{3}</math>. (Note that this is just a special case of the formula for a pyramid.) | ||
| + | ===Parallelepiped=== | ||
| + | The volume of a [[parallelepiped]] spanned by vectors <math>\bold{a} = a_1\bold{i} + a_2\bold{j} + a_3\bold{k}, \bold{b} = b_1\bold{i} + b_2\bold{j} + b_3\bold{k}, \bold{c} = c_1\bold{i} + c_2\bold{j} + c_3\bold{k}</math> is <math>|\text{det}(\begin{bmatrix} | ||
| + | c_1 & c_2 & c_3 \\ | ||
| + | a_1 & a_2 & a_3\\ | ||
| + | b_1 & b_2 & b_3 \\ | ||
| + | \end{bmatrix})|).</math> | ||
===Irregular objects=== | ===Irregular objects=== | ||
The volume of an object defined by an upper bound of <math>f(x,y,z)</math> in the Cartesian three-space can be found using a triple [[integral]]: <math>\int_{a_z}^{b_z}\int_{a_y}^{b_y}\int_{a_x}^{b_x}f(x,y,z)\text{ dx dy dz}</math>, where <math>(a_z,b_z)</math> are the bounds of <math>z</math> and similar bounds are defined for <math>x</math> and <math>y</math>. | The volume of an object defined by an upper bound of <math>f(x,y,z)</math> in the Cartesian three-space can be found using a triple [[integral]]: <math>\int_{a_z}^{b_z}\int_{a_y}^{b_y}\int_{a_x}^{b_x}f(x,y,z)\text{ dx dy dz}</math>, where <math>(a_z,b_z)</math> are the bounds of <math>z</math> and similar bounds are defined for <math>x</math> and <math>y</math>. | ||
Latest revision as of 20:24, 17 August 2023
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 of height 
and base of area 
is 
.
Pyramid
The volume of a pyramid of height and base of area is 
.
Sphere
The volume of a sphere of radius 
is 
.
Cylinder
The volume of a cylinder of height and radius is 
. (Note that this is just a special case of the formula for a prism.)
Cone
The volume of a cone of height and radius is 
. (Note that this is just a special case of the formula for a pyramid.)
Parallelepiped
The volume of a parallelepiped spanned by vectors 
is 
Irregular objects
The volume of an object defined by an upper bound of 
in the Cartesian three-space can be found using a triple integral: 
, where 
are the bounds of 
and similar bounds are defined for 
and 
.
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 
) (
