# How to determine the length of the wider base of a trapezoid from the height + shorter base length + angles?

Given a trapezoid like the one shown below, how do I determine the length of the wider base? I'm looking for a formula based method rather than drawing the shape and measuring it.

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## 1 Answer

Use the Law of Sines on the triangles (separately) on either end.

For example, on the right side of the trapezoid, drop a perpendicular from the top right vertex. Call the length of the base of the resulting right triangle $x$. The other (non-hypotenuse) side is 3/4. The bottom (interior) angle is $70^\circ$ while the top (interior) angle is $20^\circ$.

The Law of Sines says $${{3\over 4}\over \sin(70^\circ)}={x\over \sin(20^\circ)}.$$ Solve for $x$ to obtain $x={3\over 4}\cdot{\sin(20^\circ)\over \sin(70^\circ)}\approx 0.272978$.

Play the same game on the other side and you will know the total length of the base of the trapezoid.

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