📦 Volume & Surface Area

Learn and practice calculating the volume of boxes, cubes, composite shapes, and boxes with holes or cutouts.

Volume Study Guide

Learn and practice calculating the volume of boxes, cubes, composite shapes, and boxes with holes or cutouts.

Practice Topics

1. Volume of a Box

Learn how to calculate the volume of boxes, cubes, composite 3D shapes, and boxes with cutouts. Practice exercises with step-by-step explanations.

Learning Guide

1. Volume of a Box

Volume is the amount of three-dimensional space enclosed by a closed boundary. It measures how much space an object occupies, typically measured in cubic units (like

\text{cm}^3

\text{m}^3

1. Volume of a Box (Rectangular Prism)

A rectangular prism (box) has three main dimensions: length

l

, width

w

, and height

h

. The volume is found by multiplying these three dimensions together:

V = l \times w \times h

Variables:
•

V

= Volume
•

l

= length of the box
•

w

= width of the box
•

h

= height of the box

Explanation: The formula calculates the area of the base rectangle (

l \times w

) and multiplies it by the height (

h

) to extend the 2D surface into 3D space.

Volume of a Box

V = l \times w \times h

2. Cube as a Special Case

A cube is a special case of a box where all three dimensions are equal (length = width = height =

s

). Substituting

l=s

w=s

, and

h=s

into the box formula, we derive:

V = s \times s \times s = s^3

Variables:
•

V

= Volume
•

s

= side length of the cube

Volume of a Cube

V = s^3

3. Composite Shapes & Cutouts

To calculate the volume of more complex shapes, we use the properties of addition and subtraction:

Volume of Composite Shapes

V_{\text{total}} = V_1 + V_2 + \dots

When boxes are stacked or placed side-by-side, the total volume is the sum of the volumes of the individual boxes.

For shapes with hollow spaces, holes, or cutouts, we subtract the volumes:

Volume of Cutouts & Holes

V_{\text{remaining}} = V_{\text{outer}} - V_{\text{inner}}

If a smaller box or cube is cut out from a larger box, the remaining volume is the outer volume minus the inner cutout volume.

Learning Topics

Frequently Asked Questions

What is volume?

Volume is the amount of three-dimensional space occupied by an object. It is measured in cubic units, such as cubic centimeters (cm³) or cubic meters (m³).

How do you calculate the volume of a box?

The volume of a box (rectangular prism) is calculated by multiplying its length, width, and height: V = l × w × h.