Notes- Volume

Prisms

You already know that the volume of a cuboid is length by its width by its height (l × w × h).
The area of the shaded end of the cuboid (the cross section) is w × h, so you can also say that the volume of a cuboid is:
area of cross section × length

Different types of prism

This formula works for all prisms:

1. volume of a cylinder = area of circle × length
2. volume of triangular prism = area of triangle × length
3. volume of 'L'-shaped prism = area of 'L'-shape × length  
   

   Dimensional analysis

   
   The volume of a sphere is
   ⁴/₃πr³
   The surface area of a sphere is:
   4πr²
   How do we check that ⁴/₃πr³ is the formula for a volume of a shape, or that 2πr(r + L) represents a surface area? We use dimensional analysis.
   

   Dimensions of length

   In all of these formulae the letters r (radius) and L (slant height) represent lengths. We say they have 'the dimension of length'.
   The numbers are not lengths - they're just numbers. So they are called dimensionless, and we can ignore them when doing dimensional analysis (including π, which is just a number too, = 3.141...).
   

   How it works

   Here are the rules to remember for working out whether a formula is for a length, area or volume:
4. A formula with lengths occuring on their own is for a length.
5. A formula with lengths multiplied in pairs (for example, r ²) is for an area.
6. A formula with lengths cubed (for example, r³) will be for a volume.

Example 1: a volume
Let's look at the formula for the volume of a sphere.
⁴/₃πr³
If we remove all the numbers, the formula ⁴/₃πr³ becomes just r³
r³ is length × length × length
So ⁴/₃πr³ represents a volume.
Example 2: an area
Now let's look at 2πr(r + l).
By removing all the numbers from 2πr(r + l), we get just
r(r + l)
Which can be multiplied out to r² + rl
As r² and rl are both 'length x length' the formula represents an area. This makes sense as we know the formula is for the surface area of a cylinder.

Remember:
After all the numbers have been removed in a formula, you may be left with a combination of length, area and volumes. Here are the rules for simplifying these:

• length length = length
• area area = area
• volume volume = volume
• length × length = area
• length × area = volume
• 
  
• area ÷ length = length
• 
  
• volume ÷ area = length
• 
  
• volume ÷ length = area