# 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 $b$. Commented Aug 21, 2015 at 14:25
• what if the angles are known? Commented Aug 21, 2015 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 $b$.

• So all that can be found are bounds? There's no definitive formula for base a? I find that surprising. After all, we know four angles and three sides of the geometry. That seems like a lot of information. Commented Aug 21, 2015 at 14:48
• Isoceles means the angles are the same, not what their value is. Commented Aug 21, 2015 at 14:51
• i edited the question so that the angles are know. if the angles are know, then is there some formula for a? Commented Aug 21, 2015 at 14:52