Question

o is the center of the regular octagon below. find its area. round to the nearest tenth if necessary.
18
answer attempt 2 out of 2
$a = \square$ square units

Explanation:

Step1: Find central angle per triangle

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

Step2: Split triangle into right triangles

Each isosceles triangle (with height = apothem = 18) can be split into two congruent right triangles. The angle of each right triangle at the center is $\frac{45^\circ}{2}=22.5^\circ$, and one leg is the apothem ($18$), the other leg is half the side length of the octagon ($\frac{s}{2}$).

Step3: Calculate half the side length

Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$, where $\theta=22.5^\circ$, adjacent = 18:
$\tan(22.5^\circ)=\frac{\frac{s}{2}}{18}$
$\frac{s}{2}=18\times\tan(22.5^\circ)$
$\tan(22.5^\circ)=\sqrt{2}-1\approx0.4142$, so:
$\frac{s}{2}=18\times0.4142\approx7.4556$

Step4: Calculate full side length

$s=2\times7.4556\approx14.9112$

Step5: Calculate area of one triangle

Area of one isosceles triangle is $\frac{1}{2}\times s\times \text{apothem}$:
$\frac{1}{2}\times14.9112\times18\approx134.2008$

Step6: Calculate total octagon area

Multiply by 8 (number of triangles):
$8\times134.2008\approx1073.6$

Answer:

$1073.6$ square units