Math41 Home

Similarity of Shapes

Similar Figures

When one shape, enlarged or reduced by a fixed ratio, is congruent to another, the two shapes are said to have a similarity relationship. These shapes are called similar figures. In other words, if a shape is enlarged or reduced proportionally, it becomes a similar figure to the original.

For example, in the figure below,

is enlarged by a factor of

, resulting in a shape congruent to

. Thus,

and

are similar figures.

In the similar triangles

and

Points and , and , and are corresponding points.
and , and , and are corresponding sides.
and , and , and are corresponding angles.

The similarity of

and

is denoted as

, with corresponding vertices listed in the same order.

Properties of Similarity in Plane Figures

The ratio of the lengths of corresponding sides is constant.
Corresponding angles are equal.

Properties of Similarity in Solid Figures

The ratio of the lengths of corresponding edges is constant.
Corresponding faces are similar figures.

If one solid is a scaled version of another by a fixed ratio, they are said to have a similarity relationship. For example, in the figure below, rectangular prism

is an enlarged version of rectangular prism

by a factor of

. Thus,

and

are similar figures with a similarity ratio of

. Their corresponding faces are also similar figures.

Relationship Between Similarity Ratio, Area Ratio, and Volume Ratio

Area Ratio of Similar Plane Figures

The ratio of the lengths of corresponding sides in similar plane figures is called the similarity ratio.

If the similarity ratio of two plane figures is , the ratio of their areas is .

For example, two circles are always similar, with the similarity ratio equal to the ratio of their radii.

In the figure below, let the lengths of the width and height of rectangle

and

, and those of rectangle

and

, respectively. The similarity ratio of the two rectangles is

The areas of

and

are

and

, respectively. Hence, if the similarity ratio of two rectangles is

, the area ratio is

Volume Ratio of Similar Solid Figures

If the similarity ratio of two solid figures is , the ratio of their volumes is .

For example, in the figure below, let the lengths of the width, height, and depth of rectangular prism

, and

, and those of rectangular prism

, and

, respectively. The similarity ratio of the two prisms is

The volumes of

and

are

and

, respectively. Thus, if the similarity ratio of two rectangular prisms is

, the volume ratio is