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# Volume of Solids Study Guide

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Updated on Oct 3, 2011

## Introduction to Volume of Solids

Where there is matter, there is geometry.

—Johannes Kepler (1571–1630)

In this lesson, you'll discover different formulas used to determine volume, or the amount of cubic units needed to fill a three-dimensional solid.

The volume of a three-dimensional figure is the amount of space the figure occupies. Volume is always measured in cubic units, such as in.3 or cm3.

### Volume of a Cylinder

The volume of a cylinder can be found by using the formula: V = πr2h

In this formula, r is the radius of the cylinder, and h is its height.

Look at the following cylinder.

It has a height of 10 units and a radius of 3 units. Substitute these values into the volume formula:

V = πr2h V = π(3)2(10) V = π(9)(10) V = 90π

The volume is 90π cubic units. (Always remember to include the cubic units in your answer!)

#### Tip:

Questions that involve glasses usually refer to cylinders.

## Volume of a Rectangular Solid

The volume of a rectangular solid uses the following formula:

V = lwh

In this formula, l is the length of the rectangular solid, w is its width, and h is its height.

Look at the following rectangular solid.

The rectangular solid has a length of 8 units, a width of 2 units, and a height of 4 units. The volume of the rectangular solid is lwh = (8)(2)(4) = 64 cubic units.

Even though the volume of a cube can be found using the formula for the volume of a rectangular solid, there is a shortcut for the volume of a cube:

V = s3

The volume of a cube can be described as e3, where e is the length of an edge of the cube. The length, width, and height of a cube are all the same, so multiplying the length, width, and height is the same as cubing any one of those measurements.

Find practice problems and solutions for these concepts at Volume of Solids Practice Questions.

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