# Is it possible to calculate the sides of a triangle with the angles and the length from tip to base?

Is there an equation to calculate the length of a triangles sides given only the angles and the length from the tip of the triangle to its base? If so please express the equation using the following variables AX: the angle of side X L: the length from tip to base It seems like it should be possible, right? If I'm mistaken, please correct me and if there is a way to calculate the lengths without being given the lengths please tell me.

Thank you!

• What have you tried? If you know the the angles you can use the law of sines to get the proportions of the sides and you can drop an altitude to get what the proportion of the altitude. And with the actual altitude you can get the sides. Jan 25, 2019 at 16:27
• "If so please express the equation using the following variables " You said you know three angles and the height but you say to express it using the variables for one side and one angle and the height. You can't express it in a variable you don't know and using only one of the variables you do be not the other. The question you ask is entirely different than the question you describe. Jan 25, 2019 at 16:51

Let the three sides be $$a,b,c$$ (unknown variables) and the opposite angles are $$A,B,C$$ (known values) then:

$$\frac {\sin A}{a} = \frac {\sin B}b = \frac {\sin C}c$$ by the law of sines.

Let the altitude for the tip of triangle (the vertex of angle B) to the base ($$b$$) be $$h$$ (a known value). By trig definitions:

$$h = \frac {\sin A}c = \frac {\sin C}a$$.

So ....

do it.....

$$a = \frac {\sin C} h$$

$$c = \frac {\sin A} h$$

$$b = \frac a{\sin A}\sin B = \frac c{\sin C}\sin B = \frac {h\sin B}{\sin A \sin C}$$