Two rhombuses have sides with lengths of #6 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #(7pi)/8 #, what is the difference between the areas of the rhombuses?

Question

Two rhombuses have sides with lengths of #6 #.  If one rhombus has a corner with an angle of #pi/12  # and the other has a corner with an angle of #(7pi)/8  #, what is the difference between the areas of the rhombuses?

Savannah Aguilar · Accepted Answer

Difference in areas between the two rhombuses is 4.4592

Area of rhombus #= (1/2) * d_1 * d_2 or a * h#
Where #d_1 , d_2 # are the diagonals, a is the side and h is the altitude.

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

Noah Atkinson · Answer

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

The diagonals of a rhombus are perpendicular bisectors of each other, forming four right triangles within the rhombus.

Given that the sides of both rhombuses have lengths of 6, we can find the lengths of their diagonals using trigonometric functions.

For the rhombus with an angle of π/12:
The angle π/12 is the acute angle formed by the diagonal and one side of the rhombus. Using trigonometric functions, we can find that the length of the diagonal is 6 * tan(π/12) = 6 * tan(15°).

For the rhombus with an angle of (7π)/8:
The angle (7π)/8 is the acute angle formed by the diagonal and one side of the rhombus. Using trigonometric functions, we can find that the length of the diagonal is 6 * tan((7π)/8) = 6 * tan(157.5°).

Now, calculate the areas of both rhombuses using the formula mentioned earlier.

Then, find the difference between the areas of the two rhombuses.