Two sides of a rhombus form a 120º angle. The length of each side is 6 in. Explain how to find the area of the rhombus, and then calculate the area. Leave your answer in simplest radical form.
For any triangle with sides a and b,
the area is (1/2)ab sinØ , where Ø is the angle between them
Your rhombus can be split into 2 congruent triangles
so the area of a rhombus would be
absinØ,
= (6)(6)sin120° = 36(√3/2)
= 18√3 square inches
To find the area of a rhombus, we can use the formula: Area = (diagonal1 * diagonal2) / 2.
In this case, since we know the length of each side, we can determine the length of the diagonals.
Since opposite angles of a rhombus are congruent, and we are given that one angle is 120º, we can find the other angle by subtracting it from 180º:
180º - 120º = 60º
Now, we can use trigonometry to find the length of the diagonals. Since we are given the length of one side (6 in) and one angle (60º), we can find one diagonal, let's call it diagonal 1.
Using the Law of Cosines:
diagonal1^2 = 6^2 + 6^2 - 2(6)(6)cos(60º)
diagonal1^2 = 72
diagonal1 = sqrt(72)
Now, since a rhombus has two congruent diagonals, diagonal 2 will have the same length as diagonal 1.
diagonal2 = sqrt(72)
Finally, we can calculate the area:
Area = (diagonal1 * diagonal2) / 2
Area = (sqrt(72) * sqrt(72)) / 2
Area = 72 / 2
Area = 36 in^2
Therefore, the area of the rhombus is 36 square inches.