Answer:
The area is 1200cm²
Step-by-step 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:
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:
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:
Then:
• B = 75º
,
• r = 20cm
Then, the length of the side is twice x:
Now we can use the formula for the area of a dodecagon:
Then:
The area is 1200 squared cm.