Scale Models & Similarity Examples

An architect makes a model of a building with a scale of 1:18. If the ceilings in the model are 6 inches high, how high will the actual ceilings be in feet?

We need to create a proportion by writing two ratios and setting them equal to each other. We also need to make sure the scale model has tiny little people hanging out in it. Otherwise, what's the point?

The first ratio is the scale factor, which compares a measurement of the model to a measurement of the real building. The second ratio compares the height of the ceilings on the model, 6 in, to the height of the ceilings in the actual building. Since we don't know the actual height, we use the variable x.

Now we multiply both sides by 18x and solve for x.

Converting inches to feet, we get 108 ÷ 12 = 9 ft. The room will be 9 feet high.

Nikki is making a diorama of a battle scene for class, and she has little mini-trees to place in the scene. Each tree is 5 inches tall, and they're modeling trees that are 30 feet high. What is the scale factor of the model trees?

The scale is the ratio of the measurement of the model compared to the measurement of the actual tree, which can be written like this:

Nikki wants a scale with the same measurements, so she needs to do a little more work. Because the model tree is measured in inches, we convert the measurement of the actual tree into inches as well. The real tree is 30 feet tall, so we multiply 30 by 12 to get a height of 360 inches.

We can simplify the scale by dividing the numerator and denominator by 5. In other words, reduce that thang.

The scale factor of the model trees in inches is 1:72.

What is the scale factor of these two similar triangles, and what is the length of side B?

First we need to find the scale factor of the two similar triangles. There's a pair of corresponding sides (sides that are in the same position on each triangle) with measurements. The long side of the first triangle is 15 inches long, and the long side of the second triangle is 5 inches long. We can use these to find the scale factor, or the ratio of the big triangle to the little guy.

We divide both the numerator and denominator by 5 to get a scale factor of , or 3:1. That means the big guy is exactly 3 times larger than the smaller triangle.

Now we can use the scale factor to find the length of side B in the smaller triangle. Because side B corresponds to the side measuring 12 inches, we can put those together into a ratio and set it equal to the scale factor.

Multiplying both sides by B, we get 3B = 12, so B = 4 in.