 Volume of prisms and cylinders
Lesson

We have already looked at finding the amount of space enclosed by a 2D shape - this was called AREA

Well what if we have a 3D shape? How can we find the space inside of that? The space enclosed by a 3D shape is called VOLUME.

## Exploring Volume

Similar to how Area is measured in square units, Volume is measured in cubic units.

Here is $1$1 cubic unit. If we join two together we get $2$2 cubic units, add another for $3$3 cubic units and another for $4$4.  Got it? It seems easy to measure for long thin sticks.... what if the cubic units are not arranged in a line?

Here is a $2\times4$2×4 arrangement (that is $1$1 block high).  It is made up of $8$8 cubic units. If I add another row ($2$2 blocks high), it will be made up of $16$16 cubic units.  ($8$8 in the first row and $8$8 in the second) What do you think will happen if I add another row? How many cubic units now? We have $2\times4$2×4 cubic units on the bottom row, and we have 3 rows, so we have $2\times4\times3$2×4×3 cubic units = $24$24 cubic units.

Have you seen a pattern with how to work out how many cubic units there are in a rectangular solid?

## Rectangular Prisms

Rectangular solids like these are called rectangular prisms. To work out the volume of a rectangular prism we multiply the $\text{Length }\times\text{Width }$Length ×Width (of the base) and then multiply that by the number of rows, which is the $\text{Height }$Height .

Volume of Rectangular Prisms

$\text{Volume of Rectangular Prism }=\text{Length }\times\text{Width }\times\text{Height }$Volume of Rectangular Prism =Length ×Width ×Height

$V=L\times W\times H$V=L×W×H

A prism is defined as a solid geometric figure whose two end faces are similar, equal, straight and parallel.

Both of these are prisms... Prisms occur very commonly in packaging of grocery items, and finding the volume of these contributes to the design, shape and size of packaging and product - and all of these affect the price that we pay!   To experiment with how the volume of a triangular prism is affected by its base and height, or to see how the volume of a rectangular prism is affected by its base, width and height, try the following mathlet. You can vary the dimensions by moving the sliders and expose the volume by checking the checkbox.

## Volume of Any Prism

If the solid is a prism, its volume can be worked out by using the rule:

Volume of Prism

$\text{Volume }=\text{Area of Base }\times\text{Height }$Volume =Area of Base ×Height

The base is the face that is extended throughout the height. Another way of identifying the base is finding the two faces that are identical and parallel to one another.

So, in the case of a triangular prism, the volume would be found by: $\text{Area of the triangle }\times\text{height of the prism }$Area of the triangle ×height of the prism .

For a trapezoid based prism, the volume will be the $\text{area of the trapezoid }\times\text{height of the prism}$area of the trapezoid ×height of the prism .

For the volume of a complex shaped prism like this we would find the $\text{area of the base (L shape) }\times\text{height of the prism }$area of the base (L shape) ×height of the prism .

### Cylinders

A cylinder is very similar to a prism (except for the rounded face), but the volume can be found using the same process we have already learnt.

$\text{Volume of Cylinder }=\text{Area of Base }\times\text{Height of Prism }$Volume of Cylinder =Area of Base ×Height of Prism

$\text{Volume of Cylinder }=\pi r^2\times h$Volume of Cylinder =πr2×h

$\text{Volume of Cylinder }=\pi r^2h$Volume of Cylinder =πr2h

#### Example

You are at the local hardware store to buy a can of paint. After settling on one product, the salesman offers to sell you a can that is either double the height or double the radius (your choice) of the one you had decided on for double the price. Assuming all cans of paint are filled to the brim, is it worth taking up his offer?

If so, would you get more paint for each dollar if you chose the can that was double the radius or the can that was double the height?

Since the volume of a cylinder is given by the formula $\pi r^2h$πr2h, if the height doubles, the volume becomes $\pi r^2\times2h=2\pi r^2h$πr2×2h=2πr2h (ie the volume increases two-fold).

Whereas if the radius doubles, the volume becomes $\pi\left(2r\right)^2h=4\pi r^2h$π(2r)2h=4πr2h (ie the volume increases four-fold).

That is, the volume increases by double the amount when the radius is doubled compared to when the height is doubled.

To see how changes in height and radius affect the volume of a can to different extents, try the following interactive. You can vary the height and radius by moving the sliders around.

#### Practical example of Volumes

You are about to go camping for the first time and need to buy a tent. You visit the local store that sells camping gear and one tent immediately attracts your attention but it appears to be a little too small for your liking. When you ask one of the salesmen for a tent with a larger volume he points you to a tent that has a similar length but double the height and that he claims is the largest tent in store. When you ask another salesman he points you to a tent that has a similar height to your original choice but double the length and that he claims is the largest tent in store. Who is right? Can they both be right?

If you think back to the formula for the volume of a triangular prism, $\frac{1}{2}\times\text{base }\times\text{height }\times\text{length of prism }$12×base ×height ×length of prism if the prism is laid out like a tent, you will notice that when the height doubles, so too will the volume; and when the length doubles, so too will the volume. So in the case of the salesmen, both of them could be right because both of the tents they point out have the same volume.

## Units for Volume

It is probably worthwhile to remind ourselves of the units that are often used for calculations involving volume.

Units for Volume

cubic millimetres = mm3

(picture a cube with side lengths of 1 mm each - pretty small this one!)

cubic centimetres = cm3

(picture a cube with side lengths of 1 cm each - about the size of a dice)

cubic metres = m3

(picture a cube with side lengths of 1 m each - what could be this big?)

AND to convert to capacity - 1cm3 = 1mL

#### Worked Examples

##### Question 1

Find the volume of the cube shown. ##### QUESTION 2

Find the volume of the triangular prism shown. ##### QUESTION 3

Find the volume of the prism by finding the base area first. 