Problem: A regular polygon of n sides is inscribed in a circle of radius R. The area of the polygon is 3R2. Then n equals:
Answer Choices:
A. 8
B. 10
C. 12
D. 15
E. 18
Solution:
Area of the regular polygon =21 ap where a=Rcosn180 and p=n s=n \cdot 2 R \sin \dfrac{180}
∴3R2=41Rcosn180⋅2nRsinn180
∴n6=2sinn180cosn180=sinn360 where n is a positive integer equal to or greater than 3. Of the possible angles the only one whose sine is a rational number is 30∘.∴n=12. To check, note that 126=21=sin12320=sin30
or
Area of the polygon =n times the area of one triangle whose vertices are the center of the circle and two consecutive vertices of the polygon ∴3R2=n⋅21R2sinn360 or, as before, n6=sinn360.