# How do you calculate the area of a parallelogram if only two diagonals are given?

It is impossible unless an angle is given.

It is impossible to calculate the area of a parallelogram if just two diagonals are given and nothing else.

It is however possible to calculate the area if additionally, the angle between the diagonals is given.

Why is this so?

Imagine that the two diagonals are wooden sticks that are being hold together with a nail in the middle but can rotate around that nail.

Here's an example with two different parallelograms that have diagonals of the same length.

(The long diagonal is duplicated, whereas the short diagonal has kept the length but is rotated around the "nail" in the middle).

As you can see on the example, depending on the angle between the diagonals, the parallelogram that is being spread between those wooden sticks can be very different one and can have completely different areas.

In the extreme case, where the two diagonals are almost parallel, the area is very small, whereas I believe that the largest area would be if the two diagonals were orthogonal.

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You can calculate the area of a parallelogram if you are given the lengths of its two diagonals (d_1) and (d_2) using the formula:

[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 ]

This formula states that the area of a parallelogram is half the product of the lengths of its diagonals.

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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.

- A parallelogram has sides with lengths of #15 # and #8 #. If the parallelogram's area is #12 #, 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 #6 # and sides C and D have a length of # 3 #. If the angle between sides A and C is #(3 pi)/4 #, what is the area of the parallelogram?
- Two opposite sides of a parallelogram have lengths of #8 #. If one corner of the parallelogram has an angle of #pi/8 # and the parallelogram's area is #32 #, how long are the other two sides?
- Two opposite sides of a parallelogram each have a length of #25 #. If one corner of the parallelogram has an angle of #(5 pi)/12 # and the parallelogram's area is #175 #, how long are the other two sides?
- A parallelogram has sides A, B, C, and D. Sides A and B have a length of #5 # and sides C and D have a length of # 8 #. If the angle between sides A and C is #(5 pi)/8 #, what is the area of the parallelogram?

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