Two 7-sided polygons are similar. A side of the larger polygon is 6 times as long as the corresponding side of the smaller polygon. What is the ratio of the area of the larger polygon to the area of the smaller polygon?

7-side polygon is heptagon or septagon.

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"heptagon"
On en.wikipedia you have all about heptagon.

Area is:

A=3.633912444*a^2
a is side lenght

Area of smaller heptagon is:

A1=3.633912444*(a1)^2
a1 lenght side of smaller polygon

Larger side lenght is:
a2=6*a1
a2 lenght side of larger polygon

A2=3.633912444*(a2)^2
=3.633912444*(6*a1)^2
=3.633912444*36*(a1)^2

(A2/A1)=36

To find the ratio of the areas of two similar polygons, we need to know the ratio of their corresponding side lengths squared.

In this case, we are given that a side of the larger polygon is 6 times as long as the corresponding side of the smaller polygon.

Let's denote the length of the corresponding sides as x for the smaller polygon and 6x for the larger polygon.

The ratio of the corresponding side lengths squared is (6x)^2 / x^2 = 36x^2 / x^2 = 36.

Therefore, the ratio of the areas of the larger polygon to the smaller polygon is 36:1.

To find the ratio of the areas between two similar polygons, we need to use the ratio of the corresponding sides.

In this case, we are given that a side of the larger polygon is 6 times as long as the corresponding side of the smaller polygon. Let's denote the length of the corresponding side of the smaller polygon as "x". Then, the length of the corresponding side of the larger polygon is 6x.

Since the number of sides is the same for both polygons (7-sided), we know that the ratio of the areas will be proportional to the square of the ratio of the side lengths.

The ratio of the side lengths is 6x/x, which simplifies to 6. So, the ratio of the areas will be proportional to 6^2.

Therefore, the ratio of the area of the larger polygon to the area of the smaller polygon is 36:1.