A triangle has two corners with angles of 3π4 and π8. If one side of the triangle has a length of 8, what is the largest possible area of the triangle?

1 Answer
Dec 24, 2017

22.6

Explanation:

Just for me.. I prefer to work in degrees. It won't affect the area.
The area will depend on WHICH side is 8. It can be between the angles, or opposite either one.
135o and 22.5o, so the third angle is also 22.5o

That means it is an isosceles triangle with either two sides of length 8 or a base of length 8. The base must be between the two smaller angles. IF it is the length 8, the height is h=4×tan(22.5)=1.66 and the area is then:
A=12×8×1.66=6.64

If the sides are 8, the height is h=8×sin(22.5)=3.06
That makes 12 the base b=3.06tan(22.5)=7.387

and the area is then:
A=7.387×3.06=22.6

These are the only two possible areas for these conditions, so the maximum one is 22.6.

Fat and squat are larger than tall and thin! :D