Volume is the amount of space an objects takes up, this can be the amount of space a 3D shape occupies or the space that a substance (solid, liquid or gas) fills. It is measured using units such as cubic millimetres \text{(mm)}^3, cubic centimetres \text{(cm)}^3 and cubic metres \text{(m)}^3.
To find the volume of rectangular prisms we can multiply the three dimensions together. Multiplying the length by the breadth gives us the area of the base, we can then multiply this by the height to find the volume.
The volume of a rectangular prism is given by
\text{Volume} = \text{Length $\times$ Breadth $\times$ Height, or}\\ V = l \times b \times h
To find the volume of a triangular prism, we can do as we did for the rectangular prism and find the number of squares that would cover the base (area of the base) multiplied by the height.
So the volume is:
\text{Volume $=$ Area of a triangle $\times$ Height of the prism}
Since the prism can look quite different depending on the triangular face and which way it is orientated we need to be cautious about which measurements we use in our calculations.
To calculate the volume of a triangular prism we need the base and perpendicular height of the triangular face as well as the length of the prism - the distance between the two triangular faces, which is also referred to as the perpendicular height of the prism.
Given these three measurements the volume of a triangular prism can be found as follows.
\text{Volume $=$ Area of a triangle $\times$ Height of the prism, or}\\ V = Ah
For all prisms:
\text{Volume $=$ Area of the base $\times$ Height of the prism, or}\\ V=Ah
Use the applet below to explore the volume of prisms with triangular and rectangular bases.
The base area of a triangular prism where triangle serves as its base can be find by the area of triangle formula A=\dfrac12 \text{ base $\times$ height} and the base area of a rectangular prism can be find by the area of a rectangle formula A=\text{length $\times$ width}. To find the volume, this area is multiplied by the height of the prism.
Find the volume of the triangular prism shown.
Find the volume of the figure shown.
The volume of any prism is given by:
A cylinder is very similar to a prism (except for the lateral face), so the volume can be found using the same concept we have already learnt.
\displaystyle \text{Volume} | \displaystyle = | \displaystyle \text{Area of base $\times$ Height of prism} | Copy the area of a prism formula |
\displaystyle = | \displaystyle \pi r ^2 \times h | Substitute \text{Area of base $ = \pi r ^2$} | |
\displaystyle = | \displaystyle \pi r ^2 h | Simplify |
Use the applet below to explore the volume of cylinders.
The height and radius of a cylinder can be used to find its volume.
Find the volume of the cylinder rounded to two decimal places.
The volume of a cylinder is given by:
Sometimes we might know the volume of a prism but we are missing one measurement such as the the length or height. Using division, we can work backwards from the formula to find out the missing value.
For example: since a cube has equal length, width and height, if we know its volume then we can work out its side length by taking the cube root of the volume or asking ourselves "What value multiplied by itself 3 times will equal the given volume?".
A prism has a volume of 990 \text{ cm}^3.
If it has a base area of 110 \text{ cm}^2, find the height of the prism.
If we know the volume of a prism or cylinder, we can find a missing dimension by substituting the volume and given dimensions into the appropriate volume formula.