With this calculator you can calculate the area, the perimeter, the altitudes and the lengths of the diagonals of a parallelogram. The calculation path is always given as well. It can be selected to how many decimal places the calculated values should be rounded and further calculations are always performed with the rounded values.
To calculate all values of a parallelogram at least 3 input values are needed. Often, however, one does not want to calculate all values, but only a certain one and for this 2 input values are often sufficient. For example, the area can be calculated from a side length and the corresponding altitude. Therefore, it is also possible with this calculator to select that only 2 input values should be specified.
.Formulas
| Angles | α + β = 180° |
| Altitudes | ha = b ∙ sin(α) hb = a ∙ sin(β) |
| Area | A = a ∙ ha A = b ∙ hb |
| Perimeter | P = 2 ∙ (a + b) |
| Diagonal lengths |
e = a² + b² − 2 ∙ a ∙ b ∙ cos(β)
e = a² + b² + 2 ∙ a ∙ b ∙ cos(α)
f = a² + b² − 2 ∙ a ∙ b ∙ cos(α)
f = a² + b² + 2 ∙ a ∙ b ∙ cos(β)
|
| Parallelogram law | e² + f² = 2 ∙ (a² + b²) |
What is a parallelogram?
A parallelogram is a quadrilateral in which opposite sides are parallel. Due to the parallelism of the opposite sides, opposite sides also have the same length and opposite angles are equal.
In the figure above, the sides with side lengths a and c are equal in length and parallel to each other, and the sides with side lengths b and d are equal in length and parallel to each other. Furthermore, the angles α and γ are equal and the angles β and δ are equal.
Interior angles
The sum of the interior angles of a quadrilateral is always 360°. Since α = γ and β = δ holds, the sum of α and β must be 180°.
Altitudes
If you draw an altitude ha from the side with the length a in such a way that one end of the altitude lies in point C (and if necessary you also draw an extension of the line AB), then a right-angled triangle is formed. Line CB is the hypotenuse of this triangle and has length b. The angle in the right triangle at point A is either α or β. The opposite of this angle is the altitude ha.
For the acute angles in a right triangle holds:
| opposite |
| hypotenuse |
| ha |
| b |
| ha |
| b |
For angles between 0° and 180°, sin(angle) = sin(180° − angle) applies. In a parallelogram, the sum of α and β is 180°. Thus β = 180° − α applies. From this follows sin(β) = sin(180° − α) = sin(α). Thus, regardless of whether the angle in the right triangle at the corner B is α or β, the following applies:
| ha |
| b |
If you now solve the equation for ha, you get:
Equivalently, the following applies to hb:
Calculate area of a parallelogram
If you want to calculate the area of a parallelogram, you choose one side as base side, calculate the altitude belonging to this side and then multiply the length of the base side by the corresponding altitude.
The following applies to the area:
Example:
As an example, the area of a parallelogram with the side lengths a = 4 cm and b = 3 cm and the angle α = 70° is to be calculated.
With ha:
First the altitude ha is calculated.
| ha | = | b∙sin(α) |
| = | 3 cm∙sin(70°) | |
| ≈ | 2.819078 cm |
Then the area is calculated with the help of ha.
| A | = | a∙ha |
| ≈ | 4 cm∙2.819078 cm | |
| = | 11.276312 cm² |
With hb:
Now the area is to be calculated with the help of the altitude hb. First the altitude hb is calculated.
| hb | = | a∙sin(α) |
| = | 4 cm∙sin(70°) | |
| ≈ | 3.758770 cm |
Then the area is calculated with the help of hb.
| A | = | b∙ha |
| ≈ | 3 cm∙3.758770 cm | |
| = | 11.27631 cm² |
Calculate perimeter of a parallelogram
To calculate the area of a parallelogram, the sum of all 4 sides must be calculated. For a parallelogram with side lengths a and b the following applies:
The 2 can be factorised:
Example:
As an example, the parallelogram again has the side lengths a = 4 cm and b = 3 cm and for the angle α applies α = 70°. The perimeter P is to be calculated.
| P | = | 2 ∙ (a + b) |
| = | 2 ∙ (4 cm + 3 cm) | |
| = | 2 ∙ 7 cm | |
| = | 14 cm |
Lengths of the diagonals
A diagonal of a parallelogram is the distance from one corner of the parallelogram to the opposite corner. The length of the diagonal with endpoints A and C is often called e, and the length of the diagonal with endpoints B and D is called f.
The length of diagonal e can be calculated with the following equations:
and
The length of the diagonal f can be calculated with the following equations:
and
Example:
A parallelogram has the side lengths a = 4 cm and b = 3 cm and for the angle α again α = 70°. The lengths of the two diagonals e and f are to be calculated.
Calculation of e:
| e | = | a² + b² + 2 ∙ a ∙ b ∙ cos(α) |
| = | (4 cm)² + (3 cm)² + 2 ∙ 4 cm ∙ 3 cm ∙ cos(70°) | |
| = | 16 cm² + 9 cm² + 2 ∙ 4 cm ∙ 3 cm ∙ cos(70°) | |
| = | 25 cm² + 24 cm² ∙ cos(70°) | |
| ≈ | 5.76268 cm |
Calculation of f:
| f | = | a² + b² − 2 ∙ a ∙ b ∙ cos(α) |
| = | (4 cm)² + (3 cm)² − 2 ∙ 4 cm ∙ 3 cm ∙ cos(70°) | |
| = | 16 cm² + 9 cm² − 2 ∙ 4 cm ∙ 3 cm ∙ cos(70°) | |
| = | 25 cm² − 24 cm² ∙ cos(70°) | |
| ≈ | 4.09775 cm |
Parallelogram law
Once you have calculated the length of one of the two diagonals, you can also calculate the length of the other diagonal using what is called the parallelogram law:
If you have already calculated f and want to calculate e, then solve the equation for e:
And if you have already calculated e and want to calculate f next, then solve the equation for f:
Example:
The parallelogram has side lengths a = 4 cm and b = 3 cm and the angle α is 70°. The diagonal length e was calculated as in the example in the section Length of the diagonals and is approximately 5.76268 cm. Next, the length of the diagonal f is to be calculated.
| f | = | 2 ∙ (a² + b²) − e² |
| ≈ | 2 ∙ ((4 cm)² + (3 cm)²) − (5.76268 cm)² | |
| = | 2 ∙ (16 cm² + 9 cm²) − 33.2084807824 cm² | |
| = | 2 ∙ 25 cm² − 33.2084807824 cm² | |
| = | 50 cm² − 33.2084807824 cm² | |
| = | 16.7915192176 cm² | |
| ≈ | 4.09775 cm |