Area and Volume scale factor

A scale factor is a number which scales, or multiplies, some quantity. For example in this equation y=Cx, C is the scale factor for x. Scale factors are straight forward and are not as complex as other maths problems you might encounter. Once you understand the logic of solving them everything else becomes easy, there are just a few simple rules you have to understand. We shall be looking at scale factors to do with Areas and Volumes, but there are many others.

Area Scale Factor

Area scale factor can apply to any shape provided you have at least there of the quantities (areas or lengths) For example here are two triangles;

Area Scale Factor 4cm-6cm

To find the area of the large triangle, you will need to first find the linear/length scale factor as the Step 1. In general to find the scale factor we divide the large quantity with the small quantity.

Scale factor = BIG/SMALL
=6/4
=1.5

Then Step 2 we would find the Area scale factor by squaring the linear scale factor;

ASF = LSF²
=1.5² = 2.25

Step 3 we would multiple the area scale factor with the small area to find the area for the large triangle.

Area = 12 x 2.25
= 27cm2

The area for the large shape is 27cm²

NB: To find one of the lengths given the areas you will need first find the area scale factors and the square root it to find the scale factor then multiply if finding a length for a large shape or divide to find length for a small shape.

Volume Scale Factor

The same strategy applies to volume scale factors, except you have to be aware that volumes are cubic roots. For example your might have the following two bottles whose lengths are know and just one volume, and you have to find the other volume.

First step is to find the linear scale factor, remember that;

scale factor = BIG/SMALL
= 45/15
= 3


Step 2 would be to find the volume scale factor and then cubic it to find the volume scale factor.

VSF = LSF³
= 3³ = 27

Step 3 and final step would be to multiply the small volume with the big volume.

Volume = 0.7 x 27
= 18.9 litres

NB: With scale factors always remember to first find the scale factor between the two shapes or quantities and and then multiply or divide the given quantity depend on whether it’s the biggest or smallest.