Volume of Prisms

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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.

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

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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

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

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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.

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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:

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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.

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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.

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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.

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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.

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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.

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Volume of Rectangular Prism Using Formula Method

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

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Volume of Triangular Prism Using Formula Method

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

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Volume of a Cylinder Using Formula Method

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

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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.

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

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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.

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

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Metric VOLUME and Capacity Converter

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

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.

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