Answer:
Area of the regular dodecagon inscribed in a circle will be 27 square units.
Step-by-step explanation:
A regular dodecagon is the structure has twelve sides and 12 isosceles triangles inscribed in a circle as shown in the figure attached.
Since angle formed at the center by a polygon = [tex]\frac{360}{n}[/tex]
Therefore, angle at the center of a dodecagon = [tex]\frac{360}{12}[/tex] = 30°
Since one of it's vertex is (3, 0) therefore, one side of the triangle formed or radius of the circle = 3 units
Now area of a small triangle = [tex]\frac{1}{2}.(a).(b).sin\theta[/tex]
where a and b are the sides of the triangle and θ is the angle between them.
Now area of the small triangle = [tex]\frac{1}{2}.(3).(3).sin30[/tex]
= [tex]\frac{9}{4}[/tex]
Area of dodecagon = 12×area of the small triangle
= 12×[tex]\frac{9}{4}[/tex]
= 27 unit²
Therefore, area of the regular octagon is 27 square unit.
