# Given three sides of an isosceles trapezoid, find the smaller base side

I've been surprised at how challenging this problem is. Given an isosceles trapezoid, with the larger base b, the four angles, and the two equal sides c know, find the length of the shorter base a. Is the calculation even possible without knowing the height?

• Clearly not; it depends on the angle the two equal sides make with the base. If the angle is close to zero, the remaining side is close to $b-2c$; if the angle is close to $\frac{\pi}{2}$, then remaining side is close to $b$. – rogerl Aug 21 '15 at 14:25
• what if the angles are known? – nycguy92 Aug 21 '15 at 14:50

$$a = b - 2 \,c \cos \alpha$$
where $\alpha =$ angle between the big side and one of equal sides length $c$.
The height can be between $0$ and $c$, and the side $a$ can be any value between $b-2c$ and $b$.