A square with sides of length s is inscribed in an equilateral triangle with sides of length t. Find the exact ratio of the area of the equilateral triangle to the area of the square.

Question

watchmath · Answer

$\frac{6}{2+\sqrt{3}}$

anonymous · Answer

how did you get that?

anonymous · Answer

cow, remember me? i need helped, wit hthe problem you helped me yesterday

dumbcow · Answer

yeah

anonymous · Answer

hey dumbcow lol can u help me lout with my question

anonymous · Answer

http://openstudy.com/groups/mathematics#/groups/mathematics/updates/4dd9e9930ada8b0b66f142c7

dumbcow · Answer

Area of square is s^2
Area of equilateral triangle is $$\frac{\sqrt{3}}{4} t ^{2}$$ 
to get ratio we need t in terms of s
if we use the right triangle to left of square with side opposite of 60 degree angle the edge of the square of length s and adjacent side length (t-s)/2
tan 60  = opp/adj = 2s/(t-s) = sqrt(3)
solve for t
$$t = \frac{2+\sqrt{3}}{\sqrt{3}} s$$
substitute this into Area equation
$$A = \frac{\sqrt{3}}{4}*\frac{(7+4\sqrt{3})}{3} s ^{2}$$
$$A = \frac{12 + 7\sqrt{3}}{12} s ^{2}$$
divide by area of square s^2 to get ratio
$$=\frac{12+7\sqrt{3}}{12}$$

watchmath · Answer

dumbcow right, I forgot to multiplu  by two here :tan 60 = 2s/(t-s) (I did s/(t-s) )