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16 Volume

Learning Objectives

  • Understand volume as a quantity.
  • Find the volume of solids.

Volume is the amount of space an object or substance occupies. It is a three-dimensional measurement that tells us how much something can hold or how much space it takes up. Volume is often measured in cubic units (such as cubic inches, cubic centimeters, or cubic meters) because it accounts for length, width, and height.

Find the Volume of Solids

The volume of a solid shape is the amount of space the solid shape occupies.

Many common shapes have volumes that can be calculated with the following formulas. Some of these formulas can be derived from expanding the area to a volume (e.g., cubes, cylinders) but other formulas need calculus or other numerical analysis to compute volumes of (e.g., the sphere!).

Examples:

Determine the formulas for finding the volume of the following solids.

image.     image.     image

Solution: image.      image.                image

image      image     image

Solution:

image                 image                  image

Here is another summary of the volume of some basic shapes:

Examples

How Do We Measure Volume?

The formula for volume depends on the shape of the object:

  • Rectangular prism (box-shaped object):
    V=length×width×heightV = \text{length} \times \text{width} \times \text{height}     

 

  • Cylinder:
    V=πr2h     

     

    V = \pi r^2 h 

    where

    rr 

    is the radius and

    hh 

    is the height.

 

  • Sphere:
    V=43πr3V = \frac{4}{3} \pi r^3     

    where r is the radius of the sphere.

For irregular objects, volume can be found using water displacement—by measuring how much water an object pushes out when submerged.

Why is Volume Important?

Volume is used in real life in many ways, such as:

  • Cooking (measuring liquids like milk or soup)
  • Packing (fitting items into boxes or suitcases)
  • Construction (calculating materials like concrete or water tanks)

Exercises and Examples

Below are some exercises and examples of computing volume in the real wold.

Examples: Find the volumes in the following questions.

  1. A rectangular pool is 100 meters long, 80 meters wide, and 10 meters deep. What is the volume of the pool?

Solution:   image

Note the use of cubed units for volume.

Tourist buses and the Great Pyramid of Giza

 

2. The Great Pyramid at Giza is approximately 231 meters on each side of the square base and initially had a height of 147 meters. What was the volume of the Great Pyramid?

Solution:

image

1950s hot water storage tank

3. A cylindrical water heater tank has a radius of 2 feet and a height of 5 feet. What is the volume of the tank?

Solution: image

 

Football (soccer ball)

4. A soccer ball has a diameter of approximately 22 centimeters. What is the volume of a soccer ball?

Solution: image

 

 

Similar to area and surface area, to find the volume of irregular figures, we divide the given figure into shapes that we can find the volume of and add those values together to get the entire volume. This is done in the example below!

 

Examples

Grain silo, Berwick Farm

A 60-foot grain silo is in the shape of a cylinder with a semi-sphere as the roof. The radius of the sphere is 7 feet. Find the volume of the silo in bushels. This conversion is one cubic foot = 0.80 bushels.

Solution:

image

Units of Volume

There are many standard units used for volume. Here are some below:

Metric System (Used in Most Countries)

  • Cubic millimeter (mm³) – Very small volumes, like a grain of sand
  • Cubic centimeter (cm³ or cc) – Used for measuring small objects, often in medicine (e.g., medicine doses)
  • Milliliter (mL) – Used for measuring liquids (1 mL = 1 cm³)
  • Liter (L) – Common for measuring beverages, water bottles, or fuel
  • Cubic meter (m³) – Used for measuring large volumes, like a swimming pool or a room

U.S. Customary System (Used in the United States)

  • Cubic inch (in³) – Used for small objects or engine sizes
  • Cubic foot (ft³) – Used for storage, furniture, or rooms
  • Cubic yard (yd³) – Used in construction (e.g., concrete or soil)
  • Fluid ounce (fl oz) – Used for measuring drinks and cooking liquids
  • Cup, pint, quart, gallon – Common for food and beverages
  • Barrel – Used in industries like oil and beer production
Attributions
  • Content and structure adapted from RSCC Math 1410/1420 OER Team, 2022, CC BY 4.0.
  • Portions of this content adapted from Lumen Learning courseware ‘Developmental Math Emporium’ Module 8: Geometry (https://courses.lumenlearning.com/wm-developmentalemporium/). Data derived from NASA: The Cosmic Distance Scale (https://imagine.gsfc.nasa.gov/features/cosmic/earth_info.html)
  • Wikipedia: Great Pyramid of Giza (https://en.wikipedia.org/wiki/Great_Pyramid_of_Giza)
  • Image credits:
    • https://commons.wikimedia.org/wiki/File:Tourist_buses_and_the_Great_Pyramid_of_Giza.jpg
    • https://commons.wikimedia.org/wiki/File:1950s_hot_water_storage_tank_-_geograph.org.uk_-_5271620.jpg
    • https://commons.wikimedia.org/wiki/File:Football_(soccer_ball).svg
    • https://commons.wikimedia.org/wiki/File:Grain_silo,_Berwick_Farm_-_geograph.org.uk_-_2835082.jpg

License

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Mathematics for Elementary Education II Copyright © by Natalie Hobson is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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