Question 1195137
.
Find the area of the largest equilateral triangle that can be inscribed 
in a circle whose diameter is 20cm.
~~~~~~~~~~~~~~~~~~



            How the problem is worded, it shows that its composer has 
            {{{highlight(quite)}}} {{{highlight(low)}}} {{{highlight(mathematical)}}} {{{highlight(qualification)}}}.



            The correct formulation  SHOULD  NOT  speak about the largest equilateral triangle

            that can be inscribed in a circle of a given radius,  because all such triangles are congruent

            and have the same area - - - there is  NO  the  " largest "  such a triangle.



So,  I will solve the problem in that  UNIQUE  (modified)  formulation which is correct :



<pre>
    Find the area of an {{{highlight(cross(largest))}}} equilateral triangle  
    inscribed in a circle whose diameter is 20 cm.
</pre>


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<U>Solution</U>



<pre>
The sine law theorem says that if any triangle is inscribed in a circle of a radius R, then

    {{{a/sin(alpha)}}} = 2R,

where "a" is any of the three sides of the triangle and  {{{alpha}}}  is an opposite angle.



In our case, all the angles of the equilateral triangle have the same measure of 60°, so

    {{{a/sin(60^o)}}} = 2*10,

(R = 10 cm is the radius of the circle), which implies

    a = {{{2*10*(sqrt(3)/2)}}} = {{{10*sqrt(3)}}} centimeters.


Next, the area of an equilateral triangle with the side "a" is  {{{a^2*(sqrt(3)/4)}}}.


Therefore, the area of our triangle is  {{{(100*3)*(sqrt(3)/4)}}} = {{{75*sqrt(3)}}} cm^2 = 129.9038 cm^2,  approximately.    <U>ANSWER</U>
</pre>

Solved.


-----------------


To see the Sine law theorem in this formulation, &nbsp;look into the lesson


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Triangles/Law-of-sines-the-Geometric-Proof.lesson>Law of sines - the Geometric Proof</A> 


Happy learning &nbsp;(!)