The term ‘parallelogram’ is a Greek word. It is derived from the word ‘parallelogram’. The meaning of ‘parallelogram’ is any area bounded by parallel lines. Therefore, the parallelogram is a geometrical shape or a quadrilateral whose opposite pair of sides are parallel. The opposite pair of sides and angles of a parallelogram are also equal.
We know that a polygon is a closed figure in a plane that is made up of three or more line segments that intersect only at their endpoints. There are several examples of polygons like kites, rhombuses, triangles, trapezoids and parallelograms. The parallelogram is a special type of quadrilateral whose opposite pairs of sides are always parallel to each other. The opposite sides of a parallelogram are equal in length and the opposite pair of angles of a parallelogram are equal.
In mathematical language, A parallelogram is a four-sided geometric shape. It is a quadrilateral whose opposite sides are equal and are parallel to each other.
Properties of Parallelogram:
i) Opposite sides of a parallelogram are equal.
In the above parallelogram ABCD, sides AB & CD are equal and sides BC & DA are equal.
ii) Opposite angles of a parallelogram are equal.
In the above parallelogram ABCD,
∠BAD = ∠BCD
And ∠ABC = ∠ADC
iii) In a parallelogram, consecutive angles are supplementary.
In the above parallelogram ABCD,
∠BAD+∠BCD = \(180^{\circ }\)
iv) In a parallelogram, its corresponding diagonals bisect each other.
In the above parallelogram ABCD,
Let M be the point of intersection of diagonals AC & BD.
Then,
AM = MC &
BM = MD
The formula for the area of parallelogram is derived using deductive reasoning. In this process, we take the formula used to find the area of a rectangle for reference. The area of a rectangle is the product of its length ‘l’ and its width ‘w’. Similarly we can derive that the area of a parallelogram is the product of its base ‘b’ and height ‘h’. It is measured in square units like square inches, square feet, square yards etc.
The formula to find the area of a parallelogram when the values of the base and height are given is the product of its base ‘b’ and its height ‘h‘.
Area of Parallelogram (A) = b × h (Square Units)
Example 1: Find the area of a parallelogram whose base and height are 6 feet and 4 feet respectively.
Solution:
Area of parallelogram = b × h
= 6 × 4 [Replace b with 6 and h with 4]
= 24 [Simplify]
The area of parallelogram is 24 square feet
Example 2: Find the area of a parallelogram shown in figure.
Solution:
Area of parallelogram = b × h
= 6 × 15 [Replace b with 6 and h with 15]
= 90 [Simplify]
The area of parallelogram is 90 square inches.
Example 3: The total area of the parallelogram is 126 square feet. Find the base of the parallelogram if the height of the parallelogram is 14 feet.
Solution:
The height of the parallelogram is known. To find the base we use the area of a parallelogram formula to form an equation then solve.
b × h = 126 [Area of parallelogram formula]
b × 14 = 126 [Replace h with 14 ]
b = 9 [Divide both sides by 14]
The base of the parallelogram is 9 feet.
Example 4: There is a parallelogram shaped forest. The total area of the forest is 84000 square yards. There is a deer trail in the forest as shown in the figure. What is the length of the deer trail?
Solution:
The area of the parallelogram shaped forest is given. The deer trail represents the height of the parallelogram. So, use the formula for the area of the parallelogram and substitute for area and base. Then solve for the height ‘h’ which is the length of the deer trail.
A = b × h [Area of a parallelogram formula]
84000 = 600 × h [Substitute the value of Area and base of the parallelogram]
h = 140 [Divide each sides by 600]
Therefore, the deer trail is 140 yards long.
The rectangle is a special case of a parallelogram whose angles are \(90^{\circ } \)
The square is a special case of a parallelogram whose four sides are equal and every angle is \(90^{\circ } \)
If we connect the midpoints of sides of any random quadrilateral then we obtain a parallelogram. This parallelogram is called the varignon parallelogram. The area of any varignon parallelogram is half of the area of the original quadrilateral.
When the height of a parallelogram is not given, then the area of the parallelogram is calculated using trigonometric ratios.
For example, in the figure shown below,
\(\text{Area of parallelogram (A)} = \text{b}\times \text{h}\)
\(\text{Area of parallelogram (A)} = \text{b}\times \text{a}\sin\theta\)
As, \(\text{h} = \text{a}\times \sin\theta\)