VIDEO: Parallelogram: Calculate the diagonal

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Calculations on the parallelogram - how to prepare them

No matter what the task is: Always make a sketch first, in which you mark the given pieces with red paint, for example.

  1. For example, if you were to calculate the diagonal in a parallelogram, you gave the length of the two parallelogram sides and one of the four angles in the exercise.
  2. So draw a parallelogram in your sketch that should have sides of different lengths as possible. As a reminder: This is a "crooked" rectangle with opposite sides and angles of the same size. Mark the given pieces.
  3. Draw the two diagonals in your sketch, which are of different lengths. One diagonal divides the parallelogram into two general ones Triangles on.
  4. You can calculate both diagonals with the cosine theorem (collection of formulas), whereby the triangular division provides the basis for this.
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Calculate diagonals - this is how it's done

  1. First, calculate the further angle in the parallelogram, if the angle between both sides is not given. Since opposite angles are the same there, you get the missing angle by subtracting the given angle from 180 °. The sum of the angles in the parallelogram is 360 °.
  2. The law of cosines, a kind of extended Pythagoras for general triangles, allows you to calculate the side opposite the angle from two sides and the included (!) Angle. In most cases there is one for it calculator necessary.
  3. The formula for the cosine law is: c² = a² + b² - 2a * b * cos (gamma). Gamma is the angle that lies opposite side c and is enclosed by sides a and b. In this case, side c is one diagonal of the parallelogram.

Diagonals - a calculated example

A parallelogram has the two sides a = 3 cm and b = 4 cm. Of the angle between these two sides let Gamma = 70 °.

  1. Make a sketch (Fig.).
  2. Put the values ​​in the cosine block.
  3. It results for the first diagonal: c² = 9 + 16 - 24 * cos (70 °) = 25 - 8.2 = 16.8. By pulling the root you get c = 4.1 cm for the first diagonal (rounded to 2 places behind the comma).
  4. For the second diagonal, first calculate the second angle in the parallelogram. It is 110 ° (180 ° -70 °). As the sketch shows, this angle must be greater than 90 °.
  5. You can now calculate the second diagonal using the law of cosines. Note that the same triangle sides are used here, but the smaller angle that they form with each other on the second diagonal.
  6. You calculate c² = 9 + 16 - 24 * cos (110 °) = 25 + 8.2 = 33.2 and c = 5.76 cm. Note that cos (110 °) becomes negative and therefore the result of the correction term positive. You will have noticed that the larger diagonal is also opposite the larger angle - the sketch already showed it.

Special case - equilateral parallelogram

  • An equilateral parallelogram is a diamond (often also called a diamond). However, the angles in this equilateral parallelogram are not necessarily every 90 °, because then it was a square.
  • In this parallelogram, too, the two diagonals are not of the same length. Make this clear with a sketch.
  • Only in the special case of a rectangle or Square (these are also special parallelograms!) Both diagonals are of the same length.
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