Calculate Triangle From Area And Angles

Question

I'm trying to calculate triangles base on the Area and the angles. If Angle-B is 90° then the formula works, but in my case, the angle can be from 0.1° to 179.8°. The formula assumes that the angle is 90, so I was thinking that there might be something that is hidden that could work for very angle. Here is the formula:

The formula in code would be:

Height = sqrt((2 * Area) / (tan(Angle-A)));

I'm looking for the second half of the formula. Would the next part of the formula be something like this:

cos(sin(AngleB))

Answer 1

Okay, new try: If my calculations are correct, side B equals sqrt(2*area*sin(angle-B)/(sin(angle-A)*sin(angle-C))

Since Area = 1/2 * A * B * sin(c) = 1/2 * C * B * sin(a) = 1/2 * A * C * sin(b) we get:

A = 2 * area / (B * sin(c)) and using this we get:

C = sin(c) * B / sin(b) and when we place that back into the equation of area, we get:

B = sqrt(2*area*sin(angle-B)/(sin(angle-A)*sin(angle-C))

When you know one side and all the angles, calculating the other sides should be easy using normal trigonometry.

Answer 2

tziki's answer is correct, but I'd like to elaborate on how it's derived.

We start with angles and area as knowns. I'm going to use the labels in the OP's diagram for this explanation.

First we have the basic truth that the area of a triangle is half the product of its base and height: Area = base * height / 2. We want to be able to determine the relationship between base and height so that we can reduce this equation to one unknown and solve for base.

Another important thing to know is that the height of the triangle is proportional to Side-A: height = Side-A * sin(Angle-B). So knowing Side-A will give us the height.

Now we need to establish a relationship between Side-A and Side-C (the base). The most appropriate rule here is the sine law: Side-A/sin(A) = Side-C/sin(C). We re-arrange this equation to find Side-A in terms of Side-C: Side-A = Side-C * sin(A)/sin(C).

We can now insert this result into the height equation to get the formula for height in terms of Side-C only: height = Side-C * sin(A) * sin(B) / sin(C)

Using Side-C as the base in the area equation, we can now find the area in terms of Side-C only: Area = Side-C^2 * sin(A) * sin(B) / 2sin(C)

Then re-arrange this equation to find Side-C in terms of Area:

Side-C = SQRT(2 * Area * sin(C) / (sin(B) * (sin(A)))

And that gives you one side. This can be repeated to find the other sides, or you can use a different approach to find the other sides knowing this one.