o is the center of the regular octagon below. find its area. round to the nearest tenth if…

Answer:

816.8 square units

Explanation:

Step1: Find side length of octagon

A regular octagon can be divided into 8 isosceles triangles from its center. The apothem (given as 18) is the height of each triangle. The central angle for each triangle is $\frac{360^\circ}{8}=45^\circ$. Split this triangle into two right triangles: the angle at the center becomes $\frac{45^\circ}{2}=22.5^\circ$, the apothem is the adjacent side, and half the side length ($\frac{s}{2}$) is the opposite side. $\tan(22.5^\circ)=\frac{\frac{s}{2}}{18}$ $\frac{s}{2}=18\times\tan(22.5^\circ)$ $\tan(22.5^\circ)=\sqrt{2}-1\approx0.4142$ $\frac{s}{2}=18\times0.4142\approx7.4556$ $s\approx14.9112$

Step2: Find perimeter of octagon

Perimeter $P = 8\times s$ $P=8\times14.9112\approx119.2896$

Step3: Calculate area of octagon

Use the formula for area of a regular polygon: $A=\frac{1}{2}\times a\times P$, where $a$ is the apothem. $A=\frac{1}{2}\times18\times119.2896$ $A=9\times119.2896\approx1073.6064$ Wait, correction: Alternative accurate method using apothem and side relation. The formula can also be derived as $A=2(1+\sqrt{2})a^2$ for a regular octagon with apothem $a$. $A=2(1+\sqrt{2})\times18^2$ $A=2(1+1.4142)\times324$ $A=2\times2.4142\times324$ $A=4.8284\times324\approx1564.4$ Wait, correct relation: For regular octagon, apothem $a = \frac{s}{2}\times(1+\sqrt{2})$, so $s=\frac{2a}{1+\sqrt{2}}$. Then area $A=2(1+\sqrt{2})s^2/2=(1+\sqrt{2})s^2$. Substitute $s=\frac{2a}{1+\sqrt{2}}$: $A=(1+\sqrt{2})\times\left(\frac{2a}{1+\sqrt{2}}\right)^2=(1+\sqrt{2})\times\frac{4a^2}{(1+\sqrt{2})^2}=\frac{4a^2}{1+\sqrt{2}}=4a^2(\sqrt{2}-1)$ Now plug $a=18$: $A=4\times18^2\times(\sqrt{2}-1)$ $A=4\times324\times0.4142$ $A=1296\times0.4142\approx536.8$ Wait, final correct step: The standard formula for area of regular polygon is $A=\frac{1}{2}\times P\times a$. First find side length correctly: Central angle per triangle $\theta=45^\circ$, apothem $a = 18$, which is $a=\frac{s}{2\tan(\theta/2)}$ So $s=2a\tan(\theta/2)=2\times18\times\tan(22.5^\circ)=36\times0.4142=14.9112$ Perimeter $P=8\times14.9112=119.2896$ $A=\frac{1}{2}\times119.2896\times18=9\times119.2896=1073.6$ Wait, the error was in the alternative formula. The correct area using apothem is indeed $\frac{1}{2}\times P\times a$. $\tan(22.5^\circ)=\sqrt{2}-1\approx0.4142$, so $s=2\times18\times0.4142=14.9112$, perimeter $8\times14.9112=119.2896$, area $\frac{1}{2}\times119.2896\times18=1073.6$ square units.

Corrected final calculation:

Step1: Calculate half-side length

$\tan(22.5^\circ)=\frac{s/2}{18}$ $s/2=18\times(\sqrt{2}-1)\approx7.4556$

Step2: Find full side length

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

Step3: Compute octagon perimeter

$P=8\times14.9112\approx119.2896$

Step4: Calculate octagon area

$A=\frac{1}{2}\times18\times119.2896\approx1073.6$

The correct answer is approximately 1073.6 square units.