Question 869796
Let's call the missing angle {{{X}}}, and the other two angles
{{{Y=56^o}}}{{{17}}}{{{" '"}}} and {{{Z=37^o}}}{{{25}}}{{{" '"}}} .
The measure of the third angle is
{{{X=180^o-"("}}}{{{37^o}}}{{{25}}}{{{" ' "}}}{{{"+"}}}{{{56^o}}}{{{17}}}{{{" ' ) ="}}}{{{180^o-92^o}}}{{{32}}}{{{" ' ="}}}{{{86^o}}}{{{18}}}{{{" ' "}}} .
{{{X=86^o}}}{{{18}}}{{{" ' "}}} is the largest angle.
 
As customary, we call the side opposite each angle (and its length) with the same letter as the opposite angle, but lowercase instead of capital:
{{{x}}}= length (in meters) of he longest side (the one opposite largest angle X),
{{{x}}}= length (in meters) of he shortest side (the one opposite smallest angle Z), and
{{{y}}}= length (in meters) of he medium-sized side (the one opposite medium-sized angle Y).
{{{drawing(400,220,-130,80,-10,100,
triangle(-120.17,0,61.35,0,0,89.94),
locate(-2,97,X),locate(62,4,Y),locate(-125,4,Z),
locate(-30,8,x),locate(-60,45,y),locate(28,45,z)
)}}}
The sines of the angles rounded to 4 decimal places are:
{{{sin(X)=0.9979}}}
{{{sin(Y)=0.8138}}}
{{{sin(Z)=0.6076}}}
 
Law of sines states that
{{{x/sin(X)=y/sin(Y)=z/sin(Z)}}} .
Substituting the values of the sines of the angles rounded to 4 decimal places:
{{{x/0.9979=y/0.8318=z/0.6076}}} .
From those equations we can solve for {{{y}}} and {{{z}}} as functions of {{{x}}} :
{{{y/0.8318=x/0.9979}}}-->{{{y=0.8318x/0.9979}}}
{{{z/0.6076=x/0.9979}}-->{{{z=0.6076x/0.9979}}}
 
We know that the area of a triangle XYZ can be calculated from the length of two sides and the sine of the angle between them as
{{{area[XYZ]}}}={{{(1/2)*y*z*sin(X)}}}
and we know that {{{area[XYZ]}}}={{{8346}}} square meters
so substituting the values/expressions used/found above,
{{{8436=(1/2)*(0.8318x/0.9979)*(0.6076x/0.9979)*0.9979}}}
{{{8436=0.8318*0.6076*x^2/(2*0.9979)}}}
{{{2*0.9979*8436/(0.8318*0.6076)=x^2}}}
{{{x=sqrt(2*0.9979*8436/(0.8318*0.6076))}}}
{{{x=181.54}}}meters (rounded)