Calculate the length of the sides of a triangle from the area and the angle

Question

I need to find the length of the sides of a triangle. I have an angle and the area of the triangle.

I have the answer but I don't know how to figure it out so it doesn't help.

The area of the triangle is 18cm2 - the angle is 23 degrees.

Can someone explain how I can use the area and the angle to calculate the length of the sides?

Thanks

Answer 1

You obtain the 3 sides of a triangle by using this five equations (for a General triangle):

$a^2+b^2-2abcos(\alpha(a,b))=c^2$ (1)

$b^2+c^2-2bccos(\alpha(b,c))=a^2$ (2)

$a^2+c^2-2accos(\alpha(a,c))=b^2$ (3)

For the area: $A = \frac{1}{2}absin(\alpha(a,b))$ (4)

You also have the angle sum condition: $\alpha(a,b) + \alpha(b,c) +\alpha(a,c)= 180°$ (5)

From These you compute all other angles and the 3 sides of the triangle.

Answer 2

Not enough informations. Assume that $\widehat{ABC}=\widehat{A'B'C'}$ and $AB\cdot BC = A'B'\cdot B'C'$.

Then the triangles $ABC, A'B'C'$ have the same area and share an angle, but they are not necessarily congruent:

Answer 3

You are missing some critical piece of information.

To see that one angle and the area are not enough, pick all triangles that have an angle of $23$ degrees. Now shrink or dilate those triangles as necessary to get the area of $18$ cm². The results will not all have the same side lengths, as two triangles with one equal angle are not necessarily similar.