Question

o is the center of the regular nonagon below. find its perimeter. round to the nearest tenth if necessary.
6
answer attempt 1 out of 2
$p = \underline{\quad\quad\quad\quad}$ units submit answer

Explanation:

Step1: Find central angle per side

A regular nonagon has 9 sides. The central angle for one side is $\frac{360^\circ}{9} = 40^\circ$. When we split this triangle (from center to two adjacent vertices) with the apothem (the 6-unit perpendicular line), we get a right triangle with angle $\frac{40^\circ}{2}=20^\circ$.

Step2: Relate apothem to half-side length

In the right triangle, the apothem is the adjacent side to the $20^\circ$ angle, and half the side length ($s/2$) is the opposite side. Use tangent:
$\tan(20^\circ) = \frac{\frac{s}{2}}{6}$

Step3: Solve for half-side length

Rearrange to solve for $\frac{s}{2}$:
$\frac{s}{2} = 6 \times \tan(20^\circ)$
Calculate $\tan(20^\circ) \approx 0.3640$, so $\frac{s}{2} \approx 6 \times 0.3640 = 2.184$

Step4: Find full side length

Multiply by 2 to get the full side length:
$s \approx 2.184 \times 2 = 4.368$

Step5: Calculate perimeter

Perimeter $P = 9 \times s$:
$P \approx 9 \times 4.368 = 39.312$

Answer:

39.3 units