1) Prisms are 3D shapes that have the same cross-sectional shape throughout. The volume of a prism can be calculated as the area of its repeating face multiplied by its length. 2) Various examples are provided to demonstrate calculating the volume of different types of prisms such as cuboids, triangular prisms, and cylinders. 3) Other 3D shapes like cones are referred to as "pointy shapes" and their volume is calculated as the area of their base multiplied by their height. Spheres have their own unique volume formula.

1. Mr Barton’s Maths Notes
Shape and Space
6. Volume
www.mrbartonmaths.com

2. 6. Volume
The Beauty of the Prism
Good News: So long as you know what a prism is, and you remember how to work out the areas of
those 6 shapes we talked about in the last section (5. Area), you can do pretty much any volume
question without needing any more formulas!... But remember your answers are UNITS CUBED!
What is a Prism?
A Prism is a 3D object whose face is the exact same shape throughout the object.
A birthday cake is the shape of a prism if it is possible to cut it in such a way to give everyone
the exact same size piece!
prism
prism
not a prism
not a prism
prism
not a prism
prism

3. Working out the Volume of a Prism
So long as you can work out the area of the repeating face of the prism, the formula for the
volume is the same for every single one:
Volume of a Prism = Area of Repeating Face x Length
Example 1 – Cuboid
5 cm
8 cm
4 cm
FACE
Rectangle
Area = b h×
8 5× = 40cm2
Area =
Area of Repeating Face
Volume of Prism
40 4×
= 160cm3

4. Example 2 – Triangular Based Prism
11 m
6 m
5 m
15 m
Note: Don’t think you must use every measurement they give you. The 15m turned out to be
pretty useless to us!
Area of Repeating Face
FACE
Triangle
Area =
2
b h×
Area =
6
2
11×
= 33m2
Volume of Prism
33 5×
= 165m3

5. Example 3 – Cylinder
6.2 mm
3 mm
FACE
Area of Repeating Face
Circle
Area =
2
rπ ×
Area =
= 28.274… mm2
9π= ×
2
3π ×
Note: Sometimes “length” can mean “height” when you are working out the volume of the
prism. It just depends which way the repeating face is facing!
Volume of Prism
28.274... 6.2×
= 175.3mm3
(1dp)
Note: Keep this value in your calculator
and use it for the next sum. It keeps
your answer nice and accurate!

6. Example 4 – Complicated Prism Note: This is still a prism as the front face
repeats throughout the object!
FACE
Area of Repeating Face
This time it’s a bit more complicated as we cannot work
out the area of the face in one go. We must first work out
the area of the complete rectangle, and then SUBTRACT
the area of the missing circle to get our answer!
Rectangle
Area = b h×
7 5×
= 35m2
Area =
Circle
Area =
2
rπ ×
Area =
= 7.068… m2
2.25π= ×
2
51.π ×
Area of Repeating Face = 35 - 7.068…
= 27.931…
Volume of Prism
27.931 3×
= 83.8m3
(1dp)
7 m
5 m
3 m
1.5 m
Note: Try to avoid rounding in your working
out by keeping the big numbers in the
calculator, and then only round at the end!

7. Working out the Volume of Pointy Shapes
Obviously, not all 3D shapes have a repeating face. Some shapes start off with a flat face and
end up at a point. The technical name I have given to these shapes is… Pointy Shapes!
Volume of a Pointy Shape = Area of Face x Length
More Good News: Just like prisms, there is a general rule for working out the volume of all shapes
like these:
3

8. Example 4 – Cone
180 m
50 m
Area of Face
Circle
Area =
2
rπ ×
Area =
= 25,446.9… m2
8100π= ×
2
09π ×
Volume of Pointy Shape
25,446.9... 50
3
×
= 424,115 m3
(nearest whole number)
Note: Keep this value in your calculator
and use it for the next sum. It keeps
your answer nice and accurate!
FACE Diameter = 180m
Radius = 90 m

9. Example 5 – Sphere
Spheres do not have a repeating face, and they do not end in a pointy bit, so they have a rule
all to themselves, and here it is…
r Volume of a Sphere =
34
3
rπ
12 km
Volume of Sphere
34
120
3
π× ×
4
1,728,000
3
π= × ×
3
7,238,229 km=

10. Good luck with
your revision!