# General Triangles: Area, lengths and angles calculations

I have a question on General Triangles (as in non right angle). I’m trying to create a program that calculates angles and sides based on the user entering Area and some sides length or angle information.

My maths is not strong and I have searched the internet without really knowing/understanding what I'm looking for. The best I can come up with from my search the is the formula below:

$$\text{Side} = \sqrt{\frac{4}{\sqrt 3} \cdot \text{Area}}$$

I’m hoping that once I get the formula to calculate a side, I should then be able to use the Law of Sine or Cosine to calculate the remaining Angles / side lengths.

In the end I need to be able to calculate:
1. length of any side based on the area and 3 angles;

1. length of the 3 sides and the missing angle based on area and 2 angles;
2. and a few other combinations…

I’m hoping that once I get the formula to calculate a side, I should then be able to use the Law of Sine or Cosine to calculate the remaining Angles / side lengths ?

• Note that the formula you were given, namely "Side = sqrt ((4 x Area) / sqrt 3)", only works for equilateral triangles. Otherwise, this is a fairly good question. Welcome to Math.SE! – Rory Daulton Apr 1 '16 at 9:53

Your second desired calculation comes directly from the first, since you can easily find the third angle of a triangle given the other two angles: the three angles add to $180°$.

Here are some hints to find the three sides given the area and the angles.

Knowing the three angles tells you the "shape" of the triangle, up to similarity. Knowing the area tells you the "size." To put them together, just choose an arbitrary value for one of the sides, say $a_t=1$. Then you can use the Law of Cosines and/or the Law of Sines to find the other two sides. You now know a "trial triangle" that is similar to your desired triangle.

Then you can use Heron's Formula or a trigonometric formula to find the area of your trial triangle. This will almost certainly not agree with the area of your given triangle. But you know that area is proportional to the square of a side, so you can use a proportion equation to find the correct side. I.e. if the side of your trial triangle is $a_t$, the area of your trial triangle is $A_t$, and you want area $A$, you can find the corresponding correct side $a$ from

$$\left(\frac{a}{a_t}\right)^2=\frac{A}{A_t}$$

That same proportion can be used to find all three desired sides.

It is possible to put all that into one formula, but for calculation purposes I recommend you find a formula that uses the angles and area to find the ratio between the trial sides and the desired sides, then quickly use that common ratio to find the desired sides.

One shortcut is to use the final formula in the section "Using Trigonometry" in this link which finds the area given the angles and one side: you can solve that equation for the one side.

Let me know if you need more detail.