# Ratios of Lengths, Areas, and Volumes

How to find the ratios of the lengths, areas, and volumes of the similar figures: formulas, examples, and their solutions.

## Formulas

Here are two similar figures.

If the ratio of their lengths is*a*/*b*,

then the ratio of the other lengths (1D) is also*a*/*b* (= *a*^{1}/*b*^{1}).

Lengths: sides, edges, diagonals, etc.

Ratio

If the ratio of their lengths is*a*/*b*,

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

Areas: surface area, faces, etc.

If the ratio of their lengths is*a*/*b*,

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

Volumes: total volume, sliced volume, etc.

As you can see,

the dimension determines the exponents.

1D → *a*^{1}/*b*^{1}

2D → *a*^{2}/*b*^{2}

3D → *a*^{3}/*b*^{3}

## Example 1

It says these two cones are similar.

The ratio of their radii (1D) is

3/5.

(radii: plural of a radius)

So the ratio of their lateral areas (2D) is*A*/*A*' = 3^{2}/5^{2}.

3^{2} = 9

5^{2} = 25

So *A*/*A*' = 9/25.

## Example 2

It says these two cones are similar.

The ratio of their radii (1D) is

3/5.

So the ratio of their volumes (3D) is*V*/*V*' = 3^{3}/5^{3}.

3^{3} = 27

5^{3} = 125

So *V*/*V*' = 27/125.

## Example 3

To use the ratios of lengths and areas,

first find the area of the left trapezoid.

(The ratio of the heights is given: 2/3.)*b*_{1} = 3*b*_{2} = 4*h* = 2

So *A* = (1/2)(3 + 4)⋅2.

Area of a trapezoid

Cancel (1/2) and 2.

3 + 4 = 7

So *A* = 7.

The ratio of their heights (1D) is

2/3.*A* = 7

So the ratio of their areas (2D) is

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

2^{2} = 4

3^{2} = 9

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

Solve the proportion.

Then 4*A*' is equal to, 7⋅9, 63.

Proportion

Divide both sides by 4.

Then *A*' = 63/4.