How to Find Area and Perimeter of a Parallelogram with Examples
Introduction to Parallelogram Formula
A parallelogram is one of the types of quadrilaterals. A quadrilateral is a closed geometric shape which has 4 vertices, 4 sides and hence 4 angles that lie on the same plane. Sum of the interior angles of a quadrilateral measures 3600. A quadrilateral is a type of a polygon. There are various kinds of quadrilateral embracing trapezoids, parallelograms and kites. A parallelogram is one of the types of quadrilateral in which opposite sides are equal and parallel and opposite angles are equal.
Properties of Parallelogram
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Let us consider the parallelogram ABCD represented in the above figure to understand the properties of a parallelogram in a better manner. The properties of a parallelogram are listed below.
Opposites sides of a parallelogram are parallel and congruent.
i.e. AB || CD and AC || BD. Also AB = CD and AC = BD
Opposite angles of a parallelogram are congruent.
∠ABC ⩭ ∠ADC and ∠BAD ⩭ ∠BCD
Diagonals of a parallelogram bisect each other.
i.e. AC bisects BD and BD bisects AC.
Sum of any two interior adjacent angles of a parallelogram is a straight angle (i.e. it measures 1800).
If one of the interior angles of a parallelogram is a right angle, then all the interior angles are right angles.
The diagonal of a parallelogram divides the parallelogram into two congruent triangles.
i.e. If the diagonal AC is drawn for the parallelogram ABCD shown in the above figure, the diagonal divides the parallelogram into two triangles: ΔABC and ΔADC such that ΔABC ⩭ ΔADC
Important Parallelogram Formula
The Perimeter of a Parallelogram
The perimeter of a parallelogram is the measure of all sides of a parallelogram. A parallelogram is a two dimensional geometric shape. The two measurable dimensions are length and width. Since opposite sides of a parallelogram are congruent, its perimeter can be written as the sum of all the four sides in terms of length and width as:
The Perimeter of a Parallelogram = 2 (L + B)
In the above equation, ‘L’ is the length and ‘B’ is the breadth or width of the parallelogram.
Area of Parallelogram Formula
A deeper analysis of the parallelogram properties reveal that the parallelograms are made of two congruent triangles. (i.e. the diagonal of a parallelogram divides it into two congruent triangles.) Since the triangles are congruent, they have the same area. The area of a triangle is measured as half of the product of its base and height. Since the parallelogram has 2 triangles, its area is twice the area of the triangle. Therefore, the area of a parallelogram formula is equal to the product of its base and height.
Area of Parallelogram = Base x Height
Parallelogram Formula Example Problems
1. Find the area of a parallelogram whose base is 5 cm and height is 3 cm.
Solution:
Given: Length of the parallelogram / Base of the parallelogram (B) = 5 cm
Height of the parallelogram (H) = 3 cm
Area of parallelogram formula is given as:
Area = B x H
Area = 5 x 3
Area = 15 cm2
2. Determine the perimeter of a quadrilateral whose sides measure 5 cm, 4 cm, 5 cm and 4 cm taken in an order. Identify whether the given quadrilateral is a parallelogram or not. Justify your answer.
Solution:
Perimeter of a quadrilateral is the sum of all the 4 sides of the quadrilateral. So,
Perimeter = 5 + 4 + 5 + 4 = 18 cm.
The given quadrilateral is a parallelogram because its opposite sides are found to be equal. If the opposite sides are equal, obviously they will be parallel too.
Fun Quiz:
One of the most important learning outcomes of understanding the concept of parallelogram properties is that the student should be able to identify whether the given quadrilateral is a parallelogram or not.
Check whether the following quadrilaterals are parallelograms or not. Justify your answer.
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Fun Facts About Parallelogram Formula
All parallelograms are quadrilaterals and all quadrilaterals are not parallelograms.
All rectangles and squares will have the properties of parallelograms whereas all the parallelograms may not be squares or rectangles.
Square is the only example of a regular quadrilateral.
Parallelogram is a two dimensional shape and hence its volume cannot be determined.
FAQs on Parallelogram Formula for Area and Perimeter
1. What is the formula for the area of a parallelogram?
The area of a parallelogram is calculated using the formula Area = base × height (A = b × h).
- Base (b) is the length of one side.
- Height (h) is the perpendicular distance from the base to the opposite side.
- The height must form a right angle (90°) with the base.
2. How do you find the area of a parallelogram step by step?
To find the area of a parallelogram, multiply its base by its perpendicular height using A = b × h.
- Step 1: Identify the base (b).
- Step 2: Measure the perpendicular height (h).
- Step 3: Multiply base and height.
3. What is the perimeter formula of a parallelogram?
The perimeter of a parallelogram is given by P = 2(a + b), where a and b are adjacent sides.
- Add the lengths of two adjacent sides.
- Multiply the sum by 2.
4. Why is the area of a parallelogram base times height?
The area of a parallelogram is base × height because it can be rearranged into a rectangle with the same base and height.
- Cut a triangular portion from one side.
- Move it to the opposite side.
- This forms a rectangle without changing the area.
5. What is the height of a parallelogram?
The height of a parallelogram is the perpendicular distance between the base and the opposite parallel side.
- It always forms a 90° angle with the base.
- It may lie inside or outside the shape.
- It is not the slanted side length unless it is perpendicular.
6. Can you give an example of finding the area of a parallelogram?
Yes, the area is found using A = b × h.
- Given: base = 10 cm, height = 4 cm
- Area = 10 × 4
- Area = 40 cm²
7. What is the difference between a rectangle and a parallelogram?
A rectangle is a special type of parallelogram with all angles equal to 90°.
- Parallelogram: Opposite sides are parallel and equal; angles are not necessarily 90°.
- Rectangle: Opposite sides are parallel and equal; all angles are 90°.
8. How do you find the base of a parallelogram if the area and height are given?
To find the base, divide the area by the height using b = A ÷ h.
- Given: Area = 56 cm², height = 7 cm
- Base = 56 ÷ 7
- Base = 8 cm
9. What are the properties of a parallelogram?
A parallelogram has specific geometric properties related to its sides, angles, and diagonals.
- Opposite sides are parallel and equal.
- Opposite angles are equal.
- Adjacent angles are supplementary (sum = 180°).
- Diagonals bisect each other.
10. What is the formula for the area of a parallelogram using vectors?
The area of a parallelogram formed by two vectors is the magnitude of their cross product, given by Area = |a × b|.
- a and b are adjacent side vectors.
- The cross product gives a vector perpendicular to the plane.
- The magnitude represents the parallelogram’s area.
