A triangle has two corners with angles of # pi / 12 # and # pi / 6 #. If one side of the triangle has a length of #16 #, what is the largest possible area of the triangle?

1 Answer
Oct 24, 2016

Largest possible area of the triangle is #174.862#

Explanation:

As two angles of a triangle are #pi/6# and #pi/12#, the third angle is

#pi-pi/6-pi/12=(12pi-2pi-pi)/12=(9pi)/12#

All triangles with these angles are similar. As length of one side is #16#, its area will be maximum, if this side is opposite smallest angle i.e. #pi/12#.

Area of a triangle given one side #a=16# and two angles #/_A=pi/12#, #/_B=pi/6# and #/_C=(9pi)/12# is

#(a^2sinBsinC)/(2sinA)#

= #(16^2sin((9pi)/12)sin(pi/6))/(2sin(pi/12))#

= #(256xx0.7071xx1/2)/(2xx0.2588)#

= #(64xx0.7071)/0.2588#

= #174.862#