Lesson 5: More About Similarity

New Concept

When we scale a figure's dimensions by a factor of $k$, its area and volume change more dramatically. This lesson introduces the Area and Volume Principles, showing how area scales by $k^2$ and volume by $k^3$.

What’s next

Now, you'll see this principle in action through interactive examples. We'll explore how scaling dimensions affects the area and volume of various shapes.

Property

Two figures are similar if, and only if:

1. Their corresponding angles are equal, and
2. Their corresponding sides are proportional.

Examples

• A rectangle with sides 4 cm and 6 cm is similar to a rectangle with sides 8 cm and 12 cm. The scale factor is 2.

• A photograph measuring 4 inches by 6 inches is enlarged to 8 inches by 12 inches. The enlarged photo is similar to the original.

Property

If we multiply each dimension of a figure by $k$, then:

1. The new figure is similar to the original figure, and
2. The area of the new figure is $k^2$ times the area of the original figure.

Examples

• A square with a side length of 5 cm has an area of 25 cm$^2$. If you scale its dimensions by a factor of $k=3$, the new side is 15 cm and the new area is $15^2 = 225$ cm$^2$, which is $3^2 \times 25 = 9 \times 25$.

• A circular rug has a radius of 2 feet. A larger, similar rug has a radius of 6 feet. The scale factor is 3, so the area of the larger rug is $3^2=9$ times the area of the smaller one.

Property

• The ratios of all corresponding dimensions in similar objects are equal.
• The corresponding angles in similar objects are equal.
• If we scale all the dimensions of a particular object by the same number, the new object will be similar to the old one.

Examples

• A cube with 2-inch sides is similar to a cube with 6-inch sides. The ratio of their sides is $1:3$.

• Two spheres are always similar to each other. A sphere with a 5 cm radius is similar to one with a 10 cm radius.

Property

If we multiply each dimension of a three-dimensional object by $k$, then:

1. The new object is similar to the original object, and
2. The volume of the new object is $k^3$ times the volume of the original object.

Examples

• A cube with a side length of 2 cm has a volume of 8 cm$^3$. If you scale it by a factor of $k=4$, the new side is 8 cm and the new volume is $8^3 = 512$ cm$^3$, which is $4^3 \times 8 = 64 \times 8$.

• A model car is built at a $1:20$ scale. The volume of the model is $(\frac{1}{20})^3 = \frac{1}{8000}$ times the volume of the actual car.