Ratios of Lengths, Areas, and Volumes

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

If the ratio of the similar figures' 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].

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 (= a1/b1).

Lengths: sides, edges, diagonals, etc.

Ratio

If the ratio of their lengths is
a/b,

then the ratio of their areas (2D) is
a2/b2.

Areas: surface area, faces, etc.

If the ratio of their lengths is
a/b,

then the ratio of their volumes (3D) is
a3/b3.

Volumes: total volume, sliced volume, etc.

As you can see,
the dimension determines the exponents.

1D → a1/b1
2D → a2/b2
3D → a3/b3

Example 1

Two similar right cones are given below. Find the ratio of the lateral areas of the cones.

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' = 32/52.

32 = 9
52 = 25

So A/A' = 9/25.

Example 2

Two similar right cones are given below. Find the ratio of the volumes of the cones.

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' = 33/53.

33 = 27
53 = 125

So V/V' = 27/125.

Example 3

Two similar trapezoids are given below. Find the area of the right trapezoid.

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.)

b1 = 3
b2 = 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' = 22/32.

22 = 4
32 = 9

So 7/A' = 4/9.

Solve the proportion.

Then 4A' is equal to, 7⋅9, 63.

Proportion

Divide both sides by 4.

Then A' = 63/4.