Volume of Prisms

IV dispensers for drugs
Image Source: http://img.thesun.co.uk

Many jobs require a good knowledge and understanding of Volume.

Medical Personnel need to administer blood and drugs in specific volumes to patients using cylindrical injection containers and intravenous drips.

Concreters need to calculate correct volumes of material required, because concrete is expensive, and it is hard to dispose of any extra left over material.

Heating and Cooling Technicians need to calculate volumes of buildings to work out what types of systems to install.

Hairdressers need to mix chemicals together in the correct volumes when making hair treatments.

Bakers and Chefs need to measure out volumes of ingredients accurately, and know how to scale these amounts up and down for making different quantities of product.

People designing and building cars need to make sure there is enough volume inside the car for occupants and their belongings to be transported comfortably and safely, as well as ensuring the volume is correct for air bags to deploy effectively during accidents.

Water authorities in cities need to make sure that there is an adequate volume of water held in dams and reservoirs.

 
 

Definition of Volume

The Volume of a 3D Shape is the number of cubes needed to fill the inside of the shape.

In mathematics we call the 3D shape a “Prism”.

A 1cm cube is about the same size as a sugar cube.

Volume Three
Image Copyright 2013 by Passy’s World of Mathematics

 

Using some simple 3D box shapes, we can investigate Volume relationships as shown in the following examples:

Volume Four
Image Copyright 2013 by Passy’s World of Mathematics

 

Volume Five
Image Copyright 2013 by Passy’s World of Mathematics

 

Volume Six
Image Copyright 2013 by Passy’s World of Mathematics

 

From our investigation, we can see that the volume of a 3D box shape, (called a “Rectangular Prism”), is always the LENGTH x WIDTH x HEIGHT

or the Area of the bottom layer multiplied by how many stacks high the prism is.

This means that we can use the mathematical formula of V = L x W x H, (rather than drawing and counting cubes), to work out the volume of any Rectangular Prism.

 

The following video shows the cubes stacking concept for working out the formula for Rectangular Prism volume

 
 

Volume of Rectangular Prisms

Volume Seven
Image Copyright 2013 by Passy’s World of Mathematics

 

The following example shows how to calculate the volume of a box shape Rectangular Prism.

Volume Eight
Image Copyright 2013 by Passy’s World of Mathematics

 
 

Volume of Prisms – General Formula

Mathematicians have found that for any shape Prism that has two identical ends, the Volume is always the Area of the End multiplied by how long or high the prism is.

Volume Nine
Image Copyright 2013 by Passy’s World of Mathematics

Note that when the Prism is layed on its side (like the examples above), we still call the length between the identical ends the “Height”.

This is a bit confusing. Just remember that the standard position for a cylinder (or any of these Prisms), is standing up straight like a Coca Cola can, or a tin of fruit at the supermarket, and not lying on its side.

 

Area Formulas that we need to use for Prism ends are the following:

Volume Ten
Image Copyright 2013 by Passy’s World of Mathematics

 
 

Volume of a Triangular Prism

For a prism which has triangle shaped ends, we need to first find the area of the triangle using A = 1/2 x base x height of triangle.

Once we have this area, we can then multiply it by how high (or how long for sideways prisms).

Eg. V = A x H gives us the Volume of the prism.

This is shown in the following example.

Volume Eleven
Image Copyright 2013 by Passy’s World of Mathematics

 

The following video shows how to calculate the Volume of Triangular Prism.

 
 

Volume of Trapezoid Prism

For a prism which has Trapezium shaped ends, we need to first find the area of the Trapezium using A = 1/2 (top + bottom) x height of trapezium.

Once we have this area, we can then multiply it by how high (or how long for sideways prisms).

Eg. V = A x H gives us the Volume of the prism.

This is shown in the following example.

Volume Twelve
Image Copyright 2013 by Passy’s World of Mathematics

 
 

Volume of a Cylinder

For a prism which has Circle shaped ends, we need to first find the area of the Circle by Pi x Radius x Radius.

MAKE SURE THAT IF THE DIAMETER ALL THE WAY ACROSS THE CIRCLE IS GIVEN ON THE DIAGRAM, THAT YOU DIVIDE THIS DIAMETER BY 2 TO GET THE RADIUS !

Once we have this area, we can then multiply it by how high (or how long for sideways prisms).

Eg. V = A x H gives us the Volume of the cylindrical prism.

This is shown in the following example.

Volume Thirteen
Image Copyright 2013 by Passy’s World of Mathematics

 
 

Volume of an Irregular Prism

Sometimes the ends of our prism are not a simple shape like a Rectangle, Square, Triangle, Trapezium, or Circle.

This means that we cannot easily calculate the area of this odd shaped end.

For these types of prisms, the area is usually given to us in the question, and we then use this area for our V = A x H calculation.

This is shown in the following example.

Volume 14
Image Copyright 2013 by Passy’s World of Mathematics

 

However, sometime we can use “Composite Areas” method to calculate the irregular area end of the Prism.

If you are not sure about what “Composite Areas” are, then take a look at our Composite Areas lesson at the link below:

http://passyworldofmathematics.com/composite-areas/

The following video shows how to do a composite areas volume calculation for an irregular “L-shaped” prism.

 
 

Volume of Prisms Formulas

For Rectangular, Triangular, and Cylindrical Prisms, we can do calculations faster by using “all in one” formulas, where the Area x Height calculation has been combined together to give a single volume formula for the 3D shape.

The following are the “all in one” formulas we can use.

Volume 15
Image Copyright 2013 by Passy’s World of Mathematics

 
 

Volume of Rectangular Prism Using Formula Method

The following example shows how to use the “all in one” volume formula for a Rectangular Prism.

Volume 16
Image Copyright 2013 by Passy’s World of Mathematics

 
 

Volume of Triangular Prism Using Formula Method

The following example shows how to use the “all in one” volume formula for a Triangular Prism.

Volume 17
Image Copyright 2013 by Passy’s World of Mathematics

 
 

Volume of a Cylinder Using Formula Method

The following example shows how to use the “all in one” volume formula for a Cylindrical Prism.

Volume 18
Image Copyright 2013 by Passy’s World of Mathematics

 

The following video shows how to calculate the Volume of a Cylinder using the Formula Method.

 
 

This next video shows how to do a practical problem about a cylindrical storage tank.

 
 

Shown in the next video is how to ESTIMATE the volume of a cylinder if you don’t have a calculator with the exact pi value on it, and you do not want to use Pi = 3.1416 and do a difficult decimal multipication by hand.

This technique is also useful for quickly checking an answer, or for eliminating unlikely letter options on a multiple choice question.

 
 

Volume and Capacity

If we have a container filled with liquid or gas,the Volume is specified in “Capacity” units.

Capacity units are Millilitres (mL), Litres (L), Kilolitres (kL) and Megalitres (ML).

This means for Liquids and Gases in containers, we do not specify their volume in cubic centimetres or cubic meters.

The container they are in is a solid structure with Volume units of cubic centimetres or cubic meters, but the liquid or gas occupying the container has a Volume that is specified in Millilitres, or Litres.

Eg. Volume is how big the container is, but Capacity is how much the container can hold.

Container size is specified in cm or m cubed, but how much liquid or gas is in the container is specified in mL or L.

Metric Conversion Eight
Image Copyright 2013 by Passy’s World of Mathematics

 

As a typical example of a “Capacity” type Volume, consider a standard 375ml can of Coca Cola.

Metric Conversion Nine
Image Copyright 2013 by Passy’s World of Mathematics

 

MegaLitres (ML) are a very large unit of Volume that is most commonly used to specify the Volume of Water contained in large Dams and Resevoirs.

One MegaLitre = 1 million Litres.

Metric Conversion Ten
Image Copyright 2013 by Passy’s World of Mathematics

 

The following Conversion Diagram can be used for converting Capacity Units.

Metric Conversion Eleven
Image Copyright 2013 by Passy’s World of Mathematics

 
 

Metric VOLUME and Capacity Converter

Metric Conversion Twelve

The following web page has an easy to use online metric VOLUME and CAPACITY converter.

http://www.onlineconversion.com/volume.htm

 
 

Volume in the Real World

Guys pouring concrete
Image Source: http://alkispapadopoulos.com

It can be seen in the above photo that we have a rectangular prism shaped Trench, containing a cylindrical shaped Pipe.

Cement is delivered in cubic meters, and the workers would need to have calculated how much cement needed to be delivered for the job.

In this calculation they would need to have done Rectanglar Trench Volume minus the Volume of the cylinder Pipe.

If they did not do this calculation carefully and correctly, then they would either have too much cement, (which is expensive to dispose of), or not enough cement which could mean that they would not be able to complete the job on time.

 
 

Related Items

Composite Areas
Measurement Formulas
Converting Metric Measurements
Free Online Calculators and Converters

 

Subscribe

If you enjoyed this lesson, why not get a free subscription to our website.
You can then receive notifications of new pages directly to your email address.

Go to the subscribe area on the right hand sidebar, fill in your email address and then click the “Subscribe” button.

To find out exactly how free subscription works, click the following link:

How Free Subscription Works

If you would like to submit an idea for an article, or be a guest writer on our website, then please email us at the hotmail address shown in the right hand side bar of this page.

Free Gift Header

If you are a subscriber to Passy’s World of Mathematics, and would like to receive a free PowerPoint version of this lesson, that is 100% free to you as a Subscriber, then email us at the following address:

Email address image

Please state in your email that you wish to obtain the free subscriber copy of the “Volume of Prisms” Powerpoint.

Help Passy’s World Grow

Each day Passy’s World provides hundreds of people with mathematics lessons free of charge.

Help us to maintain this free service and keep it growing.

Donate any amount from $2 upwards through PayPal by clicking the PayPal image below. Thank you!





PayPal does accept Credit Cards, but you will have to supply an email address and password so that PayPal can create a PayPal account for you to process the transaction through. There will be no processing fee charged to you by this action, as PayPal deducts a fee from your donation before it reaches Passy’s World.

 
 

Enjoy,
Passy

Share
This entry was posted in Circles, Math Applications, Math in the Real World, Measurement, Measurement Formulas and tagged , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

Leave a Reply