How to Apply the Parallelogram Formula in Geometry Problems
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 Explained: Definitions, Steps & Uses
1. What are the primary formulas used to describe a parallelogram?
The main formulas for a parallelogram relate to its area and perimeter. The three most common formulas are:
Area of Parallelogram: Base × Height
Perimeter of Parallelogram: 2 × (Sum of adjacent sides)
Area using Diagonals: ½ × d₁ × d₂ × sin(θ)
2. How is the area of a parallelogram calculated using its base and height?
The most straightforward method to calculate the area of a parallelogram is by multiplying its base by its perpendicular height. The formula is Area = base × height. It is crucial to use the perpendicular height (the distance from the base to the opposite side at a 90° angle), not the length of the slanted adjacent side.
3. What is the formula for calculating the perimeter of a parallelogram?
The perimeter of a parallelogram is the total length of its boundary. Since the opposite sides of a parallelogram are equal in length, the formula is P = 2(a + b), where 'a' and 'b' are the lengths of two adjacent sides. This is because the perimeter is the sum of all four sides (a + b + a + b).
4. Why do we use the perpendicular height instead of the slanted side to find a parallelogram's area?
The area of a shape measures the two-dimensional space it covers. The formula Area = base × height is used because a parallelogram can be rearranged into a rectangle of the same area. Imagine cutting a right-angled triangle from one end of the parallelogram and moving it to the other end. This transformation creates a rectangle with the same base and height. The area of this rectangle is its length (the parallelogram's base) times its width (the parallelogram's perpendicular height). Using the slanted side would incorrectly calculate the area of a different, larger shape.
5. How can you find the area of a parallelogram if you only know the lengths of its sides and the angle between them?
If the perpendicular height is not known, you can use trigonometry. The formula for the area of a parallelogram using two adjacent sides 'a' and 'b' and the angle (θ) between them is: Area = a × b × sin(θ). This formula is especially useful in higher-level mathematics and physics problems.
6. Is there a formula to calculate the area of a parallelogram using its diagonals?
Yes, the area can also be found using the lengths of the two diagonals. If the diagonals are d₁ and d₂, and the angle between them is x, the formula is: Area = ½ × d₁ × d₂ × sin(x). This is a very useful alternative when the base and height are not provided.
7. How do the standard parallelogram formulas apply to special cases like rectangles and rhombuses?
The parallelogram formulas are the foundation for its special types:
For a Rectangle: A rectangle is a parallelogram where all angles are 90°. Here, the 'height' is simply the other side. So, the area formula base × height becomes length × width.
For a Rhombus: A rhombus is a parallelogram with all sides equal. Its diagonals bisect each other at a right angle (90°). Since sin(90°) = 1, the diagonal area formula Area = ½ × d₁ × d₂ × sin(x) simplifies to Area = ½ × d₁ × d₂.
