# Two rhombuses have sides with lengths of #2 #. 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?

≈ 1.793 square units

A rhombus has 4 equal sides and is constructed from 2 congruent isosceles triangles.

Now the area of 2 congruent triangles ( area of rhombus ) is

Difference in area = 2.828 - 1.035 = 1.793 square units

By signing up, you agree to our Terms of Service and Privacy Policy

The area of a rhombus can be calculated using the formula ( A = \frac{1}{2} d_1 d_2 ), where ( d_1 ) and ( d_2 ) are the lengths of the diagonals of the rhombus. Since the diagonals of a rhombus are perpendicular bisectors of each other, the lengths of the diagonals can be calculated using trigonometry.

For the rhombus with an angle of ( \frac{\pi}{12} ), we can calculate the length of its diagonals as follows: Let ( x ) be half the length of one diagonal, then ( \tan(\frac{\pi}{12}) = \frac{2}{x} ) which gives ( x = \frac{2}{\tan(\frac{\pi}{12})} ). Therefore, the length of one diagonal is ( 2x = \frac{4}{\tan(\frac{\pi}{12})} ). Since the diagonals are perpendicular bisectors of each other, the other diagonal is the same length.

For the rhombus with an angle of ( \frac{\pi}{4} ), both diagonals are equal in length and can be calculated using the Pythagorean theorem. Let ( y ) be half the length of one diagonal, then ( y = \frac{2}{\sqrt{2}} = \sqrt{2} ). Therefore, the length of one diagonal is ( 2y = 2\sqrt{2} ).

Now, the areas of the rhombuses can be calculated as follows: For the rhombus with angle ( \frac{\pi}{12} ), the area is ( A_1 = \frac{1}{2} \cdot \frac{4}{\tan(\frac{\pi}{12})} \cdot \frac{4}{\tan(\frac{\pi}{12})} ). For the rhombus with angle ( \frac{\pi}{4} ), the area is ( A_2 = \frac{1}{2} \cdot 2\sqrt{2} \cdot 2\sqrt{2} ).

The difference in areas is ( A_2 - A_1 ).

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- Two opposite sides of a parallelogram each have a length of #8 #. If one corner of the parallelogram has an angle of #(7 pi)/8 # and the parallelogram's area is #54 #, how long are the other two sides?
- Two rhombuses have sides with lengths of #2 #. If one rhombus has a corner with an angle of #pi/8 # and the other has a corner with an angle of #(5pi)/12 #, what is the difference between the areas of the rhombuses?
- A parallelogram has sides with lengths of #16 # and #8 #. If the parallelogram's area is #24 #, what is the length of its longest diagonal?
- A parallelogram has sides with lengths of #9 # and #8 #. If the parallelogram's area is #32 #, what is the length of its longest diagonal?
- A parallelogram has sides A, B, C, and D. Sides A and B have a length of #9 # and sides C and D have a length of # 8 #. If the angle between sides A and C is #(5 pi)/12 #, what is the area of the parallelogram?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7