In the figure, an equilateral triangle is inscribed in a circle of radius 1. The circumference of the circle is greater than the perimeter of the triangle by
A. 4π-3√3
B. 4π-(3√3)/2
C. 2π-√3
D. 2π-(3√3)/2
E. 2π-3√3

Question

In the figure, an equilateral triangle is inscribed in a circle of radius 1. The circumference of the circle is greater than the perimeter of the triangle by
A. 4π-3√3
B. 4π-(3√3)/2
C. 2π-√3
D. 2π-(3√3)/2
E.  2π-3√3

anonymous · Answer

|dw:1360235918642:dw|

agent0smith · Answer

You can use the cosine rule to find the length of the side of the equilateral triangle |dw:1360236314041:dw|
The angles are all 120 degrees because it's a circle divided into thirds.

anonymous · Answer

yes...

agent0smith · Answer

x being the side length, 
Cosine rule gives $$x^2 = r^2 + r^2 - 2r^2\cos120$$

agent0smith · Answer

Actually sine rule prob makes things easier...

anonymous · Answer

but i haven't learn sine yet..

agent0smith · Answer

Oh. Have you done the cosine rule then?

anonymous · Answer

no....

agent0smith · Answer

Maybe i'll do it the way hartnn was doing it...
|dw:1360236860124:dw|
The angle bottom left will be 30 degrees. 
$$\cos 30 = \frac{ x/2 }{ r }$$

skullpatrol · Answer

|dw:1360236024488:dw|

anonymous · Answer

x=1.732050808....

agent0smith · Answer

@skullpatrol she hasn't learned that yet.

|dw:1360237064572:dw|30 because it's an isosceles triangle, so 120+30+30 = 180.

$$\cos 30 = \frac{ x/2 }{ r } $$
$$x/2 = r \cos 30$$
$$x = 2r \cos 30$$