Thread: circumference of a parallelogram

hey guys,

i've been trying to calculate the circumference of a parallelogram by only knowing the length of the diagonals (which are 7 and 9 units).

i drew many triangles, tried to use pythagoras and some trig stuff, but failed until now...
thanks a lot for your help!

Hello, ViresArcanum!

Is there more information?
As stated, there is no unique solution.

i've been trying to calculate the perimeter of a parallelogram
by only knowing the length of the diagonals (which are 7 and 9 units).

Code:

            S                 R
            * - - - - - - - - *
           /  *           *  /
          /     * O   *     /
         /        *        / b
        /  4½ *  θ  * 3½  /
       /  *           *  /
      * - - - - - - - - *
      P        a        Q

We have parallelogram with diagonals
. . intersecting at

Law of Cosines: .

. . Hence: .

It can be shown that: .


Hence, the perimeter, , is a function of , not a constant.

Originally Posted by ViresArcanum

hey guys,

i've been trying to calculate the circumference of a parallelogram by only knowing the length of the diagonals (which are 7 and 9 units).

i drew many triangles, tried to use pythagoras and some trig stuff, but failed until now...
thanks a lot for your help!

Unfortunately that isn't possible:

The parallelogram consists of 2 congruent triangles or 2 pairs of congruent triangles. To determine one triangle unambiguously you need 3 pieces. Therefore you need the value of one more piece: an angle or the height or the length of one side.

To demonstrate how the circumference can change without changing the initial conditions (length of diagonals are constant) I've made a sketch.

Those were both really great posts.