Two similar polygons have the scale factor 5:2. The area of the larger polygon is 100. Find the area of the smaller polygon?

Question

Aaliyah Anderson · Accepted Answer

The area of the smaller polygon is:

#(2/5)^2*100 = 4/25*100 = 16#

Area changes as the square of the change in length. That is, if the lengths are double then the area is four times; if the lengths are halved, the area is one quarter, etc. In our example, the lengths of the smaller polygon are #2/5# of the lengths of the larger one, so the area is #(2/5)^2 = 4/25# times the area of the larger one.

Noah Atkinson · Answer

#area = 16#

Areas of similar figures are in the same ratio as the square of their sides.

We can write this as a proportion - the ratio of the sides is on the left, and the ratio of their areas on the right.

Note that because we are dealing with areas, the ratio of the sides is squared.

#(side_1)^2/(side_2)^2 = (area_1)/(area_2)#

#5^2/2^2 = 100/a" note:" (5/2)^2 = 5^2/2^2#

#a = (2^2xx100)/5^2#

#a = 16#

Violet Aldrich · Answer

undefined Since the scale factor between the two polygons is $5:2$, the ratio of their areas will be the square of the scale factor, which is $(5:2)^2 = 25:4$. Given that the area of the larger polygon is 100, we can set up the following proportion to find the area of the smaller polygon:

$$\frac{\text{Area of smaller polygon}}{\text{Area of larger polygon}} = \frac{4}{25}$$

$$100 \times \frac{4}{25} = 16$$

Therefore, the area of the smaller polygon is 16.