Question

o is the center of the regular octagon below. find its area. round to the nearest tenth if necessary. side length is 6.6 units.

Explanation:

Step1: Find central angle per triangle

A regular octagon can be divided into 8 congruent isosceles triangles from its center. The central angle for each triangle is:
$$\frac{360^\circ}{8} = 45^\circ$$

Step2: Split triangle into right triangles

Split one isosceles triangle into two congruent right triangles. The central angle for each right triangle is $\frac{45^\circ}{2}=22.5^\circ$, and half the side length is $\frac{6.6}{2}=3.3$ units.

Step3: Calculate apothem (height of right triangle)

Let $a$ = apothem (adjacent side to $22.5^\circ$ in the right triangle). Use $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$:
$$\tan(22.5^\circ) = \frac{3.3}{a}$$
$$a = \frac{3.3}{\tan(22.5^\circ)}$$
Since $\tan(22.5^\circ)=\sqrt{2}-1\approx0.4142$,
$$a\approx\frac{3.3}{0.4142}\approx7.967$$

Step4: Calculate perimeter of octagon

Perimeter $P$ = number of sides $\times$ side length:
$$P=8\times6.6=52.8$$

Step5: Calculate area of octagon

Use the formula for area of a regular polygon: $A=\frac{1}{2}\times a\times P$
$$A=\frac{1}{2}\times7.967\times52.8$$

Answer:

$\approx210.6$ square units