We are given that two regular polygons both have n sides. One of the polygons is inscribed, and the other is circumscribed about the circle of radius r. To find the area between two polygons, let's find the areas of each figure.
The Area of the Polygon Circumscribed About the Circle
Since any regular n-gon can be divided into n congruent isosceles triangles, let's focus on one triangle of the figure circumscribed about the circle.
In this figure the apothem is r, and to find the area we need to find the side length. First notice that the drawn triangle is made of two congruent right triangles with the legs of r and 12s, where s is the side of the polygon.
One of the acute angles of this triangle is the half of the central angle of the polygon, which is 360^(∘)n.
1/2* 360^(∘)/n=180^(∘)/n
Let's add this information to our diagram.
Next, using the tangent ratio we can write an equation and solve for s.
The side of the regular polygon is 2rtan 180^(∘)n. Finally, the area of the circumscribed circle will be n times the area of one triangle.
A_c= n*1/2* r*2rtan180^(∘)/n ⇓
A_c=nr^2tan180^(∘)/n
The Area of the Polygon Inscribed in the Circle
Now let's evaluate the area of the figure inscribed in the circle with a radius r. Again, we will focus on the one of the n triangles that form the regular polygon. Let a_i be the apothem of this polygon.
Let s_i represents the side of this polygon. Again, one of the acute angles of the right triangle is 180^(∘)n.
Let's express a_i and s_i using the cosine and the sine ratios.
sin180^(∘)/n=12s_i/r ⇒ s_i=2rsin180^(∘)/n
cos180^(∘)/n=a_i/r ⇒ a_i=rcos180^(∘)/n
Now we will use the expressions equivalent for s_i and a_i and substitute them into formula for the area of the regular polygon.
A_i=1/2* rcos180^(∘)/n* n* 2rsin180^(∘)/n ⇓ A_i=nr^2cos180^(∘)/nsin180^(∘)/n
The Area Between the Two Polygons
Finally, the area between the two polygons will be the difference between the areas of the circumscribed and the inscribed polygons. Let's substitute nr^2tan 180^(∘)n for A_c and nr^2cos 180^(∘)nsin 180^(∘)n for A_i.
