Answer to Question #158209 in Geometry for Janine Leodones

Question

The sides of two regular octagons are 3 inches and 8 inches. Find the ratio of

(a) their perimeters, (b) their radii, and (c) their areas.

Answer 1

Solution. a) Using perimeter definition

$$\frac{P_2}{P_1}=\frac{8a_2}{8a_1}=\frac{a_2}{a_1}=\frac{8}{3}$$

where a₂=8 inches and a₁=3 inches are sides of two regular octagons.

b) The radii of the inscribed and circumscribed circles are related to the side of a regular octagon by the formulas

$$R=\frac{a}{2sin22.5^0}$$

$$r=\frac{a}{2tan22.5^0}$$

where R and r are radii of regular octagon; a is side of the regular octagon. Therefore, the ratio of the radii is proportional to the ratio of the sides and is equal to

$$\frac{R_2}{R_1}=\frac{r_2}{r_1}=\frac{a_2}{a_1}=\frac{8}{3}$$

с) We use the formula for the area of a regular octagon

$$A=2a^2(1+\sqrt{2})$$

where A is area of a regular octagon; a is side of the regular octagon. Therefore

$$\frac{A_2}{A_1}=\frac{2a^2_2(1+\sqrt{2})}{2a^2_1(1+\sqrt{2})}=\frac{a^2_2}{a^2_1}=\frac{64}{9}$$

Answer. a)

$$\frac{P_2}{P_1}=\frac{8}{3}$$

b)

$$\frac{R_2}{R_1}=\frac{r_2}{r_1}=\frac{a_2}{a_1}=\frac{8}{3}$$

c)

$$\frac{A_2}{A_1}=\frac{64}{9}$$