Difference between revisions of "Volume"
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− | + | The '''volume''' of an object is a [[measure]] of the [[amount]] of [[space]] that it occupies. Note that volume only applies to three-[[dimension]]al figures. | |
− | The | + | ==Finding Volume== |
+ | This section covers the methods to find volumes of common [[Euclidean]] objects. | ||
+ | ===Prism=== | ||
+ | The volume of a [[prism]] of [[height]] <math>h</math> and base of [[area]] <math>b</math> is <math>b\cdot h</math>. | ||
+ | ===Pyramid=== | ||
+ | The volume of a [[pyramid]] of height <math>h</math> and base of area <math>b</math> is <math>\frac{bh}{3}</math>. | ||
+ | ===Sphere=== | ||
+ | The volume of a [[sphere]] of [[radius]] <math>r</math> is <math>\frac 43 r^3\pi</math>. | ||
+ | ===Cylinder=== | ||
+ | The volume of a [[cylinder]] of height <math>h</math> and radius <math>r</math> is <math>\pi r^2h</math>. (Note that this is just a special case of the formula for a prism.) | ||
+ | ===Cone=== | ||
+ | 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=== | ||
+ | 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>. | ||
− | == | + | == Problems == |
− | === Introductory | + | === Introductory === |
− | * [[ | + | *[[Center]]s of adjacent [[face]]s of a unit [[cube (geometry) | cube]] are joined to form a regular [[octahedron]]. What is the [[volume]] of this octahedron? ([[2006 AMC 10A Problems/Problem 24|Source]]) |
+ | === Intermediate === | ||
+ | *A tripod has three legs each of length <math>5</math> 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 <math>4</math> feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let <math> h </math> be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then <math> h </math> can be written in the form <math> \frac m{\sqrt{n}}, </math> where <math> m </math> and <math> n </math> are positive integers and <math> n </math> is not divisible by the square of any prime. Find <math> \lfloor m+\sqrt{n}\rfloor. </math> (The notation <math> \lfloor x\rfloor </math> denotes the greatest integer that is less than or equal to <math> x. </math>) ([[2006 AIME I Problems/Problem 14|Source]]) | ||
+ | === Olympiad === | ||
+ | ==See Also== | ||
+ | *[[Area]] | ||
+ | *[[Surface area]] | ||
+ | |||
+ | [[Category:Definition]] | ||
+ | [[Category:Geometry]] |
Latest revision as of 19:24, 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
[hide]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)