Perimeter, Area, Volume

Some of the most common and useful forms of measurement are those devoted to length, area and volume (and capacity).

We use them when calculating how far to walk or drive, the area that we may need to paint or pave or mow, and the volume of something to order, drink or use.

The perimeter is simply the distance around a shape (or an object).  The term comes from the Greek περίμετρος ‎meaning to measure around (something).  The Greek περί means around, and μετρον means measure … very simple!  Unfortunately, this does not always mean that we actually measure around something, however.  There are occasions when we must calculate the distance because we cannot physically cirumnavigate the object or because the calculation will be much more convenient.  This means that we must rely on geometry and some basic theorems and formulae about shapes.  The two that you will use most often at school will be Pythagoras’ Theorem to calculate the length of diagonal or slanted lines, and the formula for the circumference of a circle when there is an arc involved.

Areas are calculated according to some unit area (a square metre is very common).  There are a range of formulae that help us calculate the areas for common shapes.  They are all related to the area of a rectangle which we find using the formula A = lb.

There are three groups of solids that you will need to know about in order to calculate their volumes.

1. Prisms are composed of flat surfaces with two key surfaces being parallel and congruent (having the same shape and size).  Prisms have the same cross-section the entire way through.  Their areas are calculated very simply.  Once you know the area of the end (cross-section, or base), the volume is found by multiplying this by the height of the prism, i.e. V = Ah.
2. Pyramids are composed of flat surfaces also.  They have a base shape and the remaining sides all converge on a point.  The fascinating thing is that the volume of any prism is always exactly 1/3 of the volume of the prism with the same base and height!  This makes the formula very simple indeed.  If the volume of the prism is V = Ah, then the volume for the pyramid is V = Ah/3!
3. Volumes based on circles (or ellipses).  Technically a cylinder is not a prism because its surfaces are not all planes, but it bears a strong resemblance to a prism and the same volume formula (V = Ah) is used.  Since we calculate the area of the circle on its end/base using A = πr², its volume is calculated using V = πr²h.  Similarly, a cone is not a pyramid, but its volume is 1/3 that of the cylinder, so V = πr²h/3.  The volume of a sphere is calculated using V = 4πr³/3.  On these pages I will also be discussing ellipses.  Their areas are found using the simple formula V = πab, but more on that later!

How to Calculate Areas