Difference between revisions of "Volume"

Revision as of 13:21, 12 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 of height $h$ and base of area $b$ is $b\cdot h$ .

Pyramid

The volume of a pyramid of height $h$ and base of area $b$ is $\frac{bh}{3}$ .

Sphere

The volume of a sphere of radius $r$ is $\frac 43 r^3\pi$ .

Cylinder

The volume of a cylinder of height $h$ and radius $r$ is $\pi r^2h$ . (Note that this is just a special case of the formula for a prism.)

Cone

The volume of a cone of height $h$ and radius $r$ is $\frac{\pi r^2h}{3}$ . (Note that this is just a special case of the formula for a pyramid.)

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)

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 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.
+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]] 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]].
+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]] is <math>\frac 43 r^3\pi</math>, where <math>r</math> is the radius of the sphere at its widest point.
+The volume of a [[sphere]] of [[radius]] <math>r</math> is <math>\frac 43 r^3\pi</math>.
 ===Cylinder===
-The volume of a [[cylinder]] is <math>\pi r^2h</math>, where <math>h</math> is the height and <math>r</math> is the radius of the base.
+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]] is <math>\frac 13\pi r^2h</math>, where <math>h</math> is the height and <math>r</math> is the radius of the base.
+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.)
 == Problems ==

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