SOLUTION: The four sides and one diagonal of a rhombus each have sides 3√6 cm long. Find the area of the rhombus, in cm^2.

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Question 1209054: The four sides and one diagonal of a rhombus each have sides 3√6 cm long. Find the area of the rhombus, in cm^2.
Found 3 solutions by ikleyn, math_tutor2020, MathTherapy:
Answer by ikleyn(52786) About Me  (Show Source):
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The four sides and one diagonal of a rhombus each have sides 3√6 cm long.
Find the area of the rhombus, in cm^2.
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Obviously, the diagonal of 3%2Asqrt%286%29 cm  divides this rhombus in two 

equilateral congruent triangles with the sides length  a = 3%2Asqrt%286%29 cm.



The area of each such a triangle is  

    a%5E2%2A%28sqrt%283%29%2F4%29 = %283%2Asqrt%286%29%29%5E2%2A%28sqrt%283%29%2F4%29 = 9%2A6%2A%28sqrt%283%29%2F4%29 = %2854%2F4%29%2Asqrt%283%29 = %2827%2F2%29%2Asqrt%283%29.


Hence, the area of the rhombus is the doubled area of any of the two triangles  27%2Asqrt%283%29 cm^2.    ANSWER

Solved.



Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Here is a diagram to supplement the answer that tutor ikleyn has given

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
The four sides and one diagonal of a rhombus each have sides 3√6 cm long. Find the area of the rhombus, in cm^2.


Since one of its diagonals is equal to its sides, then 2 EQUILATERAL triangles will emerge, to form the rhombus

Each angle of each equilateral triangle = 180%2F3 = 60o
Area of any NON-RIGHT triangle = 1%2F2 the PRODUCT of 2 of its CONSECUTIVE sides, TIMES the sine of the INCLUDED angle 
So, AREA of one of this rhombus' 2 equilateral triangles = %281%2F2%293sqrt%286%29+%2A+3sqrt%286%29+%2A+sin+60%5Eo = %281%2F2%29%283sqrt%286%29%29%5E2%2Asin+60%5Eo 
                                                                                   --- Substituting sqrt%283%29%2F2%29 for sin 60o
As there are 2 of these equilateral triangles, AREA of both equilateral triangles/RHOMBUS = 2%2827%28sqrt%283%29%2F2%29%29%29 = cross%282%29%2827%28sqrt%283%29%2Fcross%282%29%29%29%29 =