# Ratios of Lengths, Areas, and Volumes

How to solve the ratios of lengths, areas, and volumes problem: formula, examples, and their solutions.

## Formula

The purple figure and the brown figure are similar.

If the ratio of their lengths (1D) is *a*/*b*,

then the ratio of their areas (2D) is *a*^{2}/*b*^{2},

and the ratio of their volumes (3D) is *a*^{3}/*b*^{3}.

The exponents are related to the concept's dimension:

2D → *a*^{2}/*b*^{2}

3D → *a*^{3}/*b*^{3}

## Example 1

The ratio of the radius (1D) is 3/5.

And the lateral area (*A*) is a 2D concept.

So *A*/*A*' = 3^{2}/5^{2}.

## Example 2

The ratio of the radius (1D) is 3/5.

And the volume (*V*) is a 3D concept.

So *V*/*V*' = 3^{3}/5^{3}.

## Example 3

Find the area of the little trapezoid:*A* = (1/2)⋅(3 + 4)⋅2 = 7

Area of a trapezoid

The ratio of the heights (1D) is 2/3.

And the area (*A*) is a 2D concept.

So *A*/*A*' = 2^{2}/3^{2}.

*A* = 7

So 7/*A*' = 4/9.