Two corners of a triangle have angles of π3 and π6. If one side of the triangle has a length of 4, what is the longest possible perimeter of the triangle?

1 Answer
Dec 1, 2016

The maximum perimeter is P=12+43

Explanation:

As the sum of the internal angles of a triangle is always π, if two angles are π3 and π6 the third angle equals:

ππ6π3=π2

So this is a right triangle and if H is the length of the hypotenuse,
the two legs are:

A=Hsin(π6)=H2
B=Hsin(π3)=H32

The perimeter is maximum if the side length we have is the shortest of the three, and as evidenty A<B<H then:

A=4
H=8
B=43

And the maximum perimeter is:

P=A+B+H=12+43