# Two rhombuses have sides with lengths of #7 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #(3pi)/8 #, what is the difference between the areas of the rhombuses?

Area of the rhombus with angle

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To find the difference between the areas of the two rhombuses, we first need to calculate the areas of each rhombus.

The area of a rhombus can be calculated using the formula: Area = (diagonal1 * diagonal2) / 2.

Since the lengths of the sides are given, we know that the diagonals of each rhombus will intersect at right angles, forming four congruent right-angled triangles within each rhombus.

For the rhombus with an angle of π/12, the diagonal lengths can be found using trigonometric functions. Let's denote the shorter diagonal as d1 and the longer diagonal as d2.

Using the given angle of π/12, we can find d1 and d2 using the formula: d1 = 2 * 7 * sin(π/12) d2 = 2 * 7 * cos(π/12)

Similarly, for the rhombus with an angle of (3π)/8, let's denote the shorter diagonal as d3 and the longer diagonal as d4. We can find d3 and d4 using the formula: d3 = 2 * 7 * sin((3π)/8) d4 = 2 * 7 * cos((3π)/8)

Once we have the lengths of all four diagonals for each rhombus, we can calculate their areas using the formula mentioned earlier. Then, we can find the difference between the areas of the two rhombuses by subtracting the area of the rhombus with the angle of (3π)/8 from the area of the rhombus with the angle of π/12.

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