Question

A rhombus has an angle measure of 120, and its longer diagonal has a length of 10 inches. Find the area of the rhombus.

Answer 1

Solution

              To   find   the area of   the rhombus

               In a rhombus opposite angles are equal and it contains 2 triangle (sum of degree is 2*180)
             so angles are 120, 60, 120, 60.
            and all four sides are of equal length.( = 6)
            Area   of rhombus is

                   $A = \frac{1}{2}\left ( a\times b \right )$
                    where a is the length of a diagonal
                    and b is the length of the other diagonal

              if we connect 120 degrees angles to each other will have 2 Equilateral
            it means the shorter diagonal is 6.
             and if we connect 60 degrees opposite angles to each other we will have two Rectangular.
              a² = b² + c²
              we can find longer diagonal
              36 = b² + 9
               b = 5.196
              then longer diagonal is 10.39
              Area    is

$A = \frac{1}{2}\left ( a\times b \right )$

             $A = \frac{1}{2}\left ( 10.39\times 6 \right )$
             $A = 31.17$

but   we   have other formula
        $Area = s^{2}sin(A)$
        where s is the length of any side
       and A is any interior angle
       by this formula
                 $Area = 36sin(60^{\circ})$

                  $Area = 36\times \frac{\sqrt{3}}{2}$

                 $Area = 18\sqrt{3}$