# Two rhombuses have sides with lengths of #4 #. If one rhombus has a corner with an angle of #(pi)/6 # and the other has a corner with an angle of #(3pi)/4 #, what is the difference between the areas of the rhombuses?

Difference in areas between the two rhombuses is 3.3136

Area of rhombus

Where

In this case we will use the formula Area = a * h.

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To find the difference between the areas of the two rhombuses, use the formula for the area of a rhombus, which is given by A = d1 * d2 / 2, where d1 and d2 are the lengths of the diagonals.

For a rhombus with a corner angle of π/6, the diagonals can be found using trigonometry:

- The length of one diagonal is 4 / sin(π/6) = 8.
- The length of the other diagonal is 4 / sin(π - π/6) = 4 / sin(5π/6) = 8.

Thus, for the rhombus with the corner angle of π/6, both diagonals have lengths of 8.

For a rhombus with a corner angle of 3π/4, the diagonals can also be found:

- The length of one diagonal is 4 / sin(3π/4) = 4 / (sqrt(2)/2) = 8 * sqrt(2).
- The length of the other diagonal is 4 / sin(π - 3π/4) = 4 / (sqrt(2)/2) = 8 * sqrt(2).

Thus, for the rhombus with the corner angle of 3π/4, both diagonals have lengths of 8 * sqrt(2).

Now, calculate the areas of the two rhombuses:

- For the rhombus with the corner angle of π/6, the area is (8 * 8) / 2 = 32 square units.
- For the rhombus with the corner angle of 3π/4, the area is (8 * 8 * sqrt(2)) / 2 = 32 * sqrt(2) square units.

The difference between the areas is 32 * sqrt(2) - 32 square units.

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