The 2 Pentagons Above are Similar: What is the Area of the Smaller Pentagon?

To find the area of the smaller pentagon, we need to understand how the areas of similar shapes relate to their corresponding side lengths.

When two shapes are similar, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths. Let’s denote the area of the smaller pentagon as A_small and the area of the larger pentagon as A_large. If we know the ratio of the corresponding sides of the two pentagons, we can set up the following formula:

A_small / A_large = (s_small / s_large)²

Where s_small is the length of a side of the smaller pentagon and s_large is the length of a side of the larger pentagon. If, for instance, the ratio of the sides is 1:2, the ratio of their areas would be:

A_small / A_large = (1/2)² = 1/4

This implies that the area of the smaller pentagon would be one fourth of the area of the larger pentagon. If the area of the larger pentagon is known (let’s say 100 square units), then:

A_small = A_large * (1/4) = 100 * (1/4) = 25 square units.

In summary, to find the area of the smaller pentagon, first determine the side length ratio between the two pentagons, square that ratio, and multiply it by the area of the larger pentagon. This will give you the area of the smaller pentagon.