Question

o is the center of the regular dodecagon below. find its perimeter. round to the nearest tenth if necessary.
answer attempt 1 out of 2
$p = $ units

Explanation:

Step1: Find central angle

A regular dodecagon has 12 sides, so the central angle for each side is $\frac{360^\circ}{12} = 30^\circ$.

Step2: Split triangle for side length

Draw a perpendicular from the center to a side, splitting the isosceles triangle (two sides = radius 3, vertex angle $30^\circ$) into two right triangles. The angle in the right triangle is $\frac{30^\circ}{2}=15^\circ$, and the opposite side is $\frac{s}{2}$ (half the dodecagon side $s$).

Step3: Calculate half side length

Use sine function: $\sin(15^\circ) = \frac{\frac{s}{2}}{3}$. Rearrange to get $\frac{s}{2}=3\sin(15^\circ)$.
$\sin(15^\circ)=\sin(45^\circ-30^\circ)=\sin45\cos30-\cos45\sin30=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\cdot\frac{1}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}\approx0.2588$
So $\frac{s}{2}=3\times0.2588\approx0.7764$

Step4: Find full side length

$s=2\times0.7764\approx1.5528$

Step5: Calculate perimeter

Perimeter $P=12s=12\times1.5528\approx18.6$

Answer:

18.6 units