# Two rhombuses have sides with lengths of #3 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #pi/4 #, what is the difference between the areas of the rhombuses?

Difference in areas between the two rhombuses is 4.0344

Area of rhombus

Where

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

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The area of a rhombus can be calculated using the formula: ( Area = \frac{1}{2} \times d_1 \times d_2 ), where ( d_1 ) and ( d_2 ) are the lengths of the diagonals.

For the first rhombus with an angle of ( \frac{\pi}{12} ), the diagonals are ( 3 ) and ( 3 ), because all sides are equal in a rhombus.

For the second rhombus with an angle of ( \frac{\pi}{4} ), one diagonal is ( 3 ), and to find the other diagonal, we can use trigonometry. Since the angle is ( \frac{\pi}{4} ), which is ( 45^\circ ), and the sides adjacent to the angle are equal, we can use the cosine rule:

[ \text{Adjacent side} = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2} ]

So, both diagonals of the second rhombus are ( \frac{3\sqrt{2}}{2} ).

Now, we can calculate the areas of both rhombuses.

For the first rhombus: [ Area_1 = \frac{1}{2} \times 3 \times 3 = \frac{9}{2} ]

For the second rhombus: [ Area_2 = \frac{1}{2} \times 3 \times \frac{3\sqrt{2}}{2} = \frac{9\sqrt{2}}{4} ]

The difference in their areas: [ Difference = Area_2 - Area_1 ] [ Difference = \frac{9\sqrt{2}}{4} - \frac{9}{2} ]

[ Difference = \frac{9\sqrt{2}}{4} - \frac{18}{4} ]

[ Difference = \frac{9\sqrt{2} - 18}{4} ]

So, the difference between the areas of the two rhombuses is ( \frac{9\sqrt{2} - 18}{4} ).

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