Difference between revisions of "Volume"

Revision as of 23:46, 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.

Finding Volume

This section covers the methods to find volumes of common Euclidean objects.

Prism

The volume of a prism is $Bh$ , where $B$ is the area of the base and $h$ is the height.

Pyramid

The volume of a pyramid is given by the formula $\frac13bh$ , where $b$ is the area of the base and $h$ is the height.

Sphere

The volume of a sphere is $\frac 43 r^3\pi$ , where $r$ 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 $h$ is the height $and$ r $is the radius of the base. ===Cone=== The volume of a cone is$ \frac 13\pir^2h $, where$ h $is the height$ and $r$ 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 $5$ 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 $4$ feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\frac m{\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\lfloor m+\sqrt{n}\rfloor.$ (The notation $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x.$ ) (Source)

@@ Line 1: / Line 1: @@
-The '''volume''' of an object is a [[measure]] of the [[amount]] of [[space]] that it occupies.
+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.
+==Finding Volume==
+This section covers the methods to find volumes of common [[Euclidean]] objects.
+===Prism===
+The volume of a [[prism]] is <math>Bh</math>, where <math>B</math> is the [[area]] of the base and <math>h</math> is the height.
+===Pyramid===
+The volume of a [[pyramid]] is given by the formula <math>\frac13bh</math>, where <math>b</math> is the area of the base and <math>h</math> is the [[height]].
+===Sphere===
+The volume of a [[sphere]] is <math>\frac 43 r^3\pi</math>, where <math>r</math> is the radius of the sphere at its widest point.
+===Cylinder===
+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===
+The volume of a cone is </math>\frac 13\pir^2h<math>, where </math>h<math> is the height</math> and <math>r</math> is the radius of the base.
 == Problems ==
 === Introductory  ===
-([[2006 AMC 10A Problems/Problem 24 | Source]])
+*[[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]]
-{{stub}}
 [[Category:Definition]]
 [[Category:Geometry]]

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