A parallelogram is a two-dimensional geometrical, non-intersecting quadrilateral with two pairs of equal and parallel sides. In a parallelogram, the opposite sides are equal in length, and the opposite angles are also equivalent. A parallelogram will always have four sides, four vertices, and four angles as it is a type of quadrilateral. The three-dimensional version has the faces in the shape of a parallelogram and is called a parallelepiped. In this article, we will see how to calculate the area, perimeter, properties, and special cases of a parallelogram.
Area of a Parallelogram
The area is defined as the region enclosed within given boundaries. Suppose we have a parallelogram with side lengths given by m, n, and X, Y are the angles between the adjacent sides; then the area is given by the following formula:
- Area of parallelogram = m * n sin (X) = m * n sin (Y).
There is another way to calculate the area. If we know the height of the parallelogram, that is, a perpendicular has been dropped from one vertex to the opposite side, then the following formula can be used:
- Area of a parallelogram = base * height.
Finally, if we know the length of the diagonals, we can apply another formula. Say the diagonals are denoted by d1 and d2 while the angle of intersection is given by V, then the area of parallelogram is given by
- Area of a parallelogram = ½ * d1 * d2 * sin (V).
Perimeter of a Parallelogram
The perimeter is defined as the total length of the boundaries of a shape. Suppose we have a parallelogram with side lengths m and n, then the perimeter is given by
Perimeter of a parallelogram = m + n + m + n (As the opposite sides are equal) = 2m + 2n = 2 (m + n)
Types of Parallelograms
If a few constraints are applied, we can find several special cases of parallelograms. A few are listed below:
- Rectangle: A parallelogram that has all four angles measuring 90 degrees with opposite sides equal and parallel.
- Square: A parallelogram that has all four angles measuring 90 degrees with all sides equal and parallel.
- Rhombus: If all the sides are equal, then such a parallelogram is called a rhombus.
Properties of Parallelogram
- The opposite sides are equal and parallel. By definition, the opposite sides never intersect.
- The sum of all the interior angles of a parallelogram is equal to 360 degrees.
- Adjacent angles are supplementary; that is, they sum up to 180 degrees.
- The opposite angles and sides are congruent.
- Two congruent triangles are formed by the diagonals of the parallelogram.
- The diagonals are bisectors of each other.
- If one of the interior angles is right-angled, then all the others will also be 90 degrees.
- According to the parallelogram law, the sum of squares of all sides is equal to the sum of squares of the two diagonals.
- Four triangles of equal areas are formed by both the diagonals when they intersect.
There are many questions that can be based on parallelograms requiring the manipulation of the formulas as well as improving the speed and accuracy in computation. The best way to gain these skills and combat confusion is by joining an online mathematical platform such as Cuemath. At Cuemath, the math experts use several methods that ensure the installation of robust concepts within students. The techniques range from using different resources to providing cues to a child etc. The end goal for them is to help kids dispel their fear of mathematics while making the learning experience enjoyable. Hopefully, this article gives you an insight into parallelograms!