A parallelogram is a two-dimensional geometrical shape, whose sides are parallel with each other. It is made up of four sides, where the pair of parallel sides are equal in length. Also, the opposite angles of a parallelogram are equal to each other. The area of parallelogram depends on the base and height of it.
In geometry, you must have learned about many 2D shapes and sizes such as circle, square, rectangle, rhombus, etc. All of these shapes have a different set of properties. Also, the area and perimeter formulas of these shapes vary with each other, used to solve many problems. Let us learn here the definition, formulas and properties of a parallelogram.
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A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length and the opposite angles are equal in measure.
In the figure above, you can see, ABCD is a parallelogram, where AB//CD and AD//BC.
Also, AB = CD and AD = BC
And, ∠A = ∠C & ∠B = ∠D
The formula for area and parameter of a parallelogram covered here in this section. Students can use these formulas and solve problems based on them.
Area of Parallelogram
Area of a parallelogram is the region occupied by it in a two-dimensional plane. Below is the formula to find the parallelogram area:
Area = Base × Height
In the above figure, //gramABCD, Area is given by;
|Area = a b sin A = b a sin B|
where a is the slant length of the side of //gramABCD and b is the base.
Check here: Area of a Parallelogram Formula
Perimeter of Parallelogram
The perimeter of any shape is the total distance of the covered around the shape or its total length of any shape. Similarly, the perimeter of a parallelogram is the total distance of the boundaries of the parallelogram. To calculate the perimeter value we have to know the values of its length and breadth. The parallelogram has its opposite sides equal in length. Therefore, the formula of the perimeter could be written as;
|Perimeter = 2 (a+b)|
Properties of Parallelogram
If a quadrilateral has a pair of parallel opposite sides, then it’s a special polygon called Parallelogram. The properties of a parallelogram are as follows:
- The opposite sides are congruent.
- The opposite angles are congruent.
- The consecutive angles are supplementary.
- If anyone of the angles is a right angle, then all the other angles will be right.
- The two diagonals bisect each other.
- Each diagonal bisects the parallelogram into two congruent triangles.
- The diagonals separate it into congruent.
Types of Parallelogram
There are mainly four types of Parallelogram depending on various factors. The factors which distinguish between all of these different types of parallelogram are angles, sides etc.
- In a parallelogram, say PQRS
- If PQ = QR = RS = SP are the equal sides, then it’s a rhombus. All the properties are the same for rhombus as for parallelogram.
- Other two special types of a parallelogram are:
Is Square a Parallelogram?
Square could be considered as a parallelogram since the opposite sides are parallel to each other and the diagonals of the square bisect each other.
Is Rectangle a Parallelogram?
Theorem 1: Parallelograms on the same base and between the same parallel sides are equal in area.
Proof: Two parallelograms ABCD and ABEF, on the same base DC and between the same parallel line AB and FC.
To prove that area (ABCD) = area (ABEF).
Parallelogram ABCD and rectangle ABML are on the same base and between the same parallels AB and LC.
area of parallelogram ABCD = area of parallelogram ABML
We know that area of a rectangle = length x breadth
Therefore, area of parallelogram ABCD = AB x AL
Hence, the area of a parallelogram is the product of any base of it and the corresponding altitude.
In ∆ADF and ∆BCE,
AD=BC (∴ABCD is a parallelogram ∴ AD=BC)
AF=BE (∴ABEF is a parallelogram ∴AF=BE)
∠ADF=∠BCE (Corresponding Angles)
∠AFD=∠BEC (Corresponding Angles)
∠DAF =∠CBE (Angle Sum Property)
∆ADE ≅ ∆BCF (From SAS-rule)
Area(ADF) = Area(BCE) (By congruence area axiom)
Area(ABCD)=Area(ABED) + Area(BCE)
Hence, the area of parallelograms on the same base and between the same parallel sides is equal.
Corollary: A parallelogram and a rectangle on the same base and between the same parallels are equal in area.
Proof: Since a rectangle is also a parallelogram so, the result is a direct consequence of the above theorem.
Theorem: The area of a parallelogram is the product of its base and the corresponding altitude.
Given: In a parallelogram ABCD, AB is the base.
To prove that Area(||gmABCD) = AB×AL
Construction: Complete the rectangle ALMB by Drawing BM perpendicular to CD.
Examples of Parallelogram
|Example- Find the area of a parallelogram whose base is 5 cm and height is 8 cm.
Solution- Given, Base = 5 cm and Height = 8 cm.
We know, Area = Base x Height
Area = 5 × 8
Area = 40 Sq.cm
Example: Find the area of a parallelogram having length of diagonals to be 10 and 22 cm and an intersecting angle to be 65 degrees.
Solution: We know that the diagonals of a parallelogram bisect each other, hence the length of half the diagonal will be 5 and 11 cm.
The angle opposite to the side b comes out to be 180 – 65 = 115°
We use the law of cosines to calculate the base of the parallelogram –
b² = 5² + 11² – 2(11)(5)cos(115°)
b² = 25 + 121 – 110(-.422)
b² = 192.48
b = 13.87 cm.
After finding the base we need to calculate the height of the given parallelogram.
To find the height we have to calculate the value of θ, so we use sine law
5/sin(θ) = b/sin(115)
Now we extend the base and draw in the height of the figure and denote it as ‘h’.
The right-angled triangle (marked with red line) has the Hypotenuse to be 22 cm and Perpendicular to be h.
sin θ = h/22
h = 7.184 cm
Area = base × height
A = 13.87 × 7.184
A = 99.645 sq.cm
Hope this discussion has made all your doubts clear regarding Parallelograms and their properties. Register with BYJU’S to learn more about quadrilaterals and other maths concepts.