Answer:
The area is 1200cm²
Explanation:
To solve this problem, we need to find the side length. We can divide the dodecagon in isosceles triangles
The angle A we can calculate it, because the dodecagon is composed by 12 triangles like this. Since the sum of all angles A add up to a whole circle:
[tex]\angle A=\frac{360º}{12}=30º[/tex]
Since each triangle is an isosceles triangle, the two angles at the bottom are the same. Also, the sum of the internal angles of a triangle is 180º. Then:
[tex]\begin{gathered} A+B+B=180º \\ 30º+2B=180º \\ B=\frac{180º-30º}{2} \end{gathered}[/tex][tex]B=75º[/tex]
And finally, we can calculate x, which is half of the length of each side, using trigonometric relationships. In this case, we can use cosine:
[tex]\cos(B)=\frac{x}{r}[/tex]
Then:
• B = 75º
,
• r = 20cm
[tex]\begin{gathered} \cos(75º)=\frac{x}{20cm} \\ x\approx5.176cm \end{gathered}[/tex]
Then, the length of the side is twice x:
[tex]L=2\cdot5.18cm=10.35cm[/tex]
Now we can use the formula for the area of a dodecagon:
[tex]A=3(2+\sqrt{3})\cdot L^2[/tex]
Then:
[tex]A=3(2+\sqrt{3})(10.35)^2=1200cm^2[/tex]
The area is 1200 squared cm.
