A quadrilateral is a closed shape formed by the intersection of four line segments. A parallelogram is formed when the opposite sides of a quadrilateral are parallel and of equal length. The length of the continuous line produced by a parallelogram’s boundary is its perimeter. The perimeter of a parallelogram formula can be articulated as \(P = 2\left( {a + b} \right){\rm{units}}.\)

The examples of parallelogram involve rhombus, square, rectangles, etc. In this article, we will discuss about the perimeter of a parallelogram formula in detail. Scroll down to read more!

Perimeter: Overview

Perimeter is the total length of the boundary of a plane figure.

What is Parallelogram?

A quadrilateral is a closed shape formed by the junction of four line segments. A parallelogram is formed when the opposite sides of a quadrilateral are parallel and are of equal length.
Some of the examples of a parallelogram are as follows:
1. Square
2. Rectangle
3. Rhombus

What is the Perimeter of Parallelogram?

The length of the continuous line produced by a parallelogram’s boundary is its perimeter.

In the case of the parallelogram, its perimeter will be obtained by adding the lengths of all its four sides.

So, if the lengths of the sides of the parallelogram shown in the figure above is given by \(a\) and \(b,\) then the perimeter \(\left( P \right)\) of a parallelogram with sides is \(P = 2\left( {a + b} \right)\,{\rm{units}}.\)

Properties of a Parallelogram

Properties of a parallelogram are mentioned below:

1. A parallelogram’s opposite sides are congruent.
2. A parallelogram’s opposite angles are congruent.
3. A parallelogram’s consecutive angles are supplementary.
4. The diagonals of a parallelogram bisect each other.
5. A parallelogram’s each diagonal divides the parallelogram into two triangles of equal area.

What is the Formula for Perimeter of a Parallelogram?

The perimeter of a parallelogram is equal to the sum of all its sides since it is the total length of its outline. It is easy to calculate the perimeter of the lengths of the three sides are given. However, we can also calculate the perimeter of a parallelogram if (a) one side and two diagonals and (b) base, height and the measure of one angle are given.

Hence, we can calculate the perimeter of a parallelogram in the following three cases:
1. Measures of all four sides are given
2. Measures of one side and two diagonals are given
3. Measures of a base, height and angle are given

How to Calculate Perimeter of a Parallelogram?

1. The Perimeter of a Parallelogram when measures of all four sides are given

We know that the perimeter of a parallelogram is the sum of the lengths of all its sides.
The opposite sides of a parallelogram are equal. Let \(a\) and \(b\) are sides of the parallelogram.
The perimeter \(\left( P \right)\) of a parallelogram with sides is,
\(P = \left( {a + b + a + b} \right)\)
\(P = 2\left( {a + b} \right)\,{\rm{units}}{\rm{.}}\)

2. The Perimeter of a Parallelogram when measures of one side and two diagonals are given

We were given side \(AD = BC = a\) and \(AC = y,\,BD = x.\)
Let another side \(AB = CD = b\)
By using the Law of Cosines,
For \(\Delta ABD,\,{x^2} = {a^2} + {b^2} – 2ab\,\cos \,\angle DAB\)…..(i)
By using the Law of Cosines,
For \(\Delta ADC,\,{y^2} = {a^2} + {b^2} – 2ab\,\cos \,\angle ADC\)…..(ii)
By adding the equation (i) and equation(ii).
\({x^2} + {y^2} = {a^2} + {b^2} – 2\,ab\,\cos \,\angle DAB + {a^2} + {b^2} – 2ab\,\cos \,\angle ADC.\)
\({x^2} + {y^2} = 2{a^2} + 2{b^2} – 2ab\,\left( {\cos \angle DAB + \cos \,\angle ADC} \right)\,\)…(iii)
From the property of the parallelogram, two adjacent angles of a parallelogram are supplementary.
\(\angle DAB + \angle ADC = {180^{\rm{o}}}\)
\(\angle DAB = {180^{\rm{o}}} – \angle ADC\)
Applying cosine on both sides
\(\cos \,\angle DAB = \cos \left( {{{180}^{\rm{o}}} – \angle ADC} \right) = \, – \cos \left( {\angle ADC} \right)\)
Substitute the value of \(\cos \,\angle DAB\) in equation (iii)
\({x^2} + {y^2} = 2{a^2} + 2{b^2} – 2ab\left( { – \cos \left( {\angle ADC} \right) + \cos \,\angle ADC} \right)\)
\({x^2} + {y^2} = 2{a^2} + 2{b^2}\) (Relationship between Sides and Diagonals.)
Find the unknown side \(b\) from the above equation.
\({b^2} = \frac{{\left( {{x^2} + {y^2} – 2{a^2}} \right)}}{2}\)
\(\therefore b = \sqrt {\frac{{\left( {{x^2} + {y^2} – 2{a^2}} \right)}}{2}} \)
We know that
The perimeter \(\left( P \right)\) of a parallelogram with two sides is \(P = 2\left( {a + b} \right)\)
\(P = 2\left( {a + \sqrt {\frac{{\left( {{x^2} + {y^2} – 2{a^2}} \right)}}{2}} } \right)\)
\( \Rightarrow P = 2a + 2\left( {\sqrt {\frac{{\left( {{x^2} + {y^2} – 2{a^2}} \right)}}{2}} } \right)\)
\( \Rightarrow P = 2a + \left( {\sqrt {\frac{{4 \times \left( {{x^2} + {y^2} – 2{a^2}} \right)}}{2}} } \right) = 2a + \left( {\sqrt {2\left( {{x^2} + {y^2} – 2{a^2}} \right)} } \right)\)
\( \Rightarrow P = 2a + \left( {\sqrt {\left( {2{x^2} + 2{y^2} – 4{a^2}} \right)} } \right).\)

3. The Perimeter of a Parallelogram when measures of a base, height and angle are given

We were given side \(AB = CD = a\) and height \(BE = h,\,\,\angle BCE = \theta \) and other sides \(BC = AD = b.\)
From \(\Delta BEC,\)
\(\sin \,\theta = \frac{{{\rm{opposite}}}}{{{\rm{hypotenuse}}}} = \frac{h}{b}\)
\(\therefore b = \frac{h}{{\sin \,\theta }}\)
We know, the perimeter \(\left( P \right)\) of a parallelogram with sides is, \(P = 2\left( {a + b} \right)\)
\(P = 2\left( {a + \frac{h}{{\sin \,\theta }}} \right)\)
\(P = \left( {2a + \frac{{2h}}{{\sin \,\theta }}} \right)\)
\(\sin \,\theta = \sin \,\left( {{{180}^{\rm{o}}} – \theta } \right)\) for any \(\theta \)
\(\theta \) doesn’t need to be a specific angle of the parallelogram. Because any two adjacent angles of a parallelogram are supplementary, it can be any vertex angle.

Importance and Uses of Perimeter

Below we have provided the importance and uses of perimeter:

1. Perimeter of a parallelogram is an important parameter in geometry. It is used in a lot of applications in mathematics.
2. If you have a garden whose shape is of a parallelogram, we may calculate the cost of its fencing after calculating its perimeter.
3. Perimeter of a parallelogram has a lot of applications in the field of surveying, flooring estimates, architectural, mechanical engineering, etc.

Example of Perimeter Parallelogram

Your home has a fenced yard whose shape is that of a parallelogram, as shown in the figure below. If the yard is \(50\) feet by \(40\) feet, find the perimeter of the yard.

In this case, the length of the fence of the yard is the perimeter.
The perimeter of a parallelogram \(P = 2\left( {{\rm{length}} + {\rm{breadth}}} \right)\,{\rm{units}}{\rm{.}}\)
The perimeter of a yard \(P = 2\left( {50 + 40} \right)\)
\( \Rightarrow P = 2\left( {90} \right)\)
\( \Rightarrow P = 180\)
Therefore, the perimeter of the yard is \(180\,{\rm{feet}}{\rm{.}}\)

Solved Examples on Perimeter of a Parallelogram Formula

Q.1. Find the length of another side of the parallelogram whose base is \(10\,{\rm{cm}}\) and the perimeter is \(50\,{\rm{cm}}\).
Ans: From the given information, we get, base \(\left( b \right) = 10\,{\rm{cm,}}\) Perimeter \(\left( P \right) = 50\,{\rm{cm,}}\) Length \(\left( a \right) = ?\)
The perimeter of a parallelogram is given by \(P = 2\left( {a + b} \right)\,{\rm{units}}\)
\( \Rightarrow 50 = 2\left( {a + 10} \right)\)
\( \Rightarrow 50 = 2a + 20\)
\( \Rightarrow 2a = 50 – 20\)
\( \Rightarrow 2a = 30\)
\( \Rightarrow a = \frac{{30}}{2}\)
\( \Rightarrow a = 15\)
Therefore, \(15\,{\rm{cm}}\) is the length of another side of the parallelogram.

Q.2. Find the perimeter of a parallelogram whose adjacent sides are 5 units and 9 units.
Ans: From the given length \(\left( a \right) = 5\,{\rm{units,}}\) base \(\left( b \right) = 9\,{\rm{units,}}\) perimeter \(\left( P \right) = ?\)
The perimeter of a parallelogram \(P = 2\left( {a + b} \right){\rm{units}}.\)
\( \Rightarrow P = 2\left( {5 + 9} \right)\)
\( \Rightarrow P = 2 \times 14\)
\( \Rightarrow P = 28\)
Therefore, \(28\,{\rm{units}}\) is the perimeter of a parallelogram.

Q.3. Find the perimeter of a parallelogram whose length is \(14\,{\rm{cm}}\) and width is \(11\,{\rm{cm}}\).
Ans: From the given length \(\left( a \right) = 14\,{\rm{cm,}}\) width \(\left( b \right) = 11\,{\rm{cm,}}\) perimeter \(\left( P \right) = ?\)
The perimeter of a parallelogram \(P = 2\left( {a + b} \right)\,{\rm{units}}\)
\( \Rightarrow P = 2\left( {14 + 11} \right)\)
\( \Rightarrow P = 2 \times 25\)
\( \Rightarrow P = 50\)
Therefore, \(50\,{\rm{cm}}\) is the perimeter of a parallelogram.

Q.4. What is the perimeter of a parallelogram with the side length \(6\,{\rm{cm}}\) and diagonals \(8\,{\rm{cm}}\) and \(10\,{\rm{cm}}\). Round your answer to two decimals.
Ans: From the given, \({\rm{length}}\,\left( a \right) = 6\,{\rm{cm}},\,{\rm{diagona}}{{\rm{l}}_1}\left( x \right) = 8\,{\rm{cm}},\,{\rm{diagona}}{{\rm{l}}_2}\left( y \right) = 10\,{\rm{cm}}\)
The formula to find the perimeter of a parallelogram with one side and two diagonals,
\(P = 2a + \left( {\sqrt {2{x^2} + 2{y^2} – 4{a^2}} } \right)\)
\( \Rightarrow P = 2 \times 6 + \left( {\sqrt {2{{\left( 8 \right)}^2} + 2{{\left( {10} \right)}^2} – 4{{\left( 6 \right)}^2}} } \right)\)
\( \Rightarrow P = 12 + \left( {\sqrt {128 + 200 – 144} } \right)\)
\( \Rightarrow P = 12 + \left( {\sqrt {328 – 144} } \right)\)
\( \Rightarrow P = 12 + \left( {\sqrt {184} } \right)\)
\( \Rightarrow P = 12 + 13.5646…\)
\( \Rightarrow P = 12 + 13.56\) (Upto two decimal places)
\( \Rightarrow P = 25.56\)
Therefore, the perimeter of a given parallelogram is \(25.56\,{\rm{cm}}.\)

Q.5. What is the perimeter of a parallelogram where one of its sides is 12 yards, its corresponding height is 8 yards, and one of the vertex angles is 30 degrees.
Ans: From the given length \(\left( a \right) = 12\,{\rm{yards}},\) height \(\left( h \right) = 8\,{\rm{yards}}\) vertex angle \(\left( \theta \right) = {30^{\rm{o}}}\)
\(P = \left( {2a + \frac{{2h}}{{\sin \,\theta }}} \right)\)
\( \Rightarrow P = \left( {2 \times 12 + \frac{{2 \times 8}}{{\sin \,{{30}^{\rm{o}}}}}} \right)\)
\( \Rightarrow P = \left( {24 + \frac{{16}}{{\frac{1}{2}}}} \right)\)
\( \Rightarrow P = \left( {24 + 32} \right)\)
\( \Rightarrow P = 56\,{\rm{yards}}\)
Therefore, \(56\,{\rm{yards}}\) is the perimeter of a parallelogram.

Q.6. What is the perimeter of a parallelogram where one of its sides is 14 yards, its corresponding height is 10 yards, and one of the vertex angles is 90 degrees.
Ans: From the given length \(\left( a \right) = 14\,{\rm{yards}},\) height \(\left( h \right) = 10\,{\rm{yards}}\) vertex angle \(\left( \theta \right) = {45^{\rm{o}}}.\)
\(P = \left( {2a + \frac{{2h}}{{\sin \,\theta }}} \right)\)
\( \Rightarrow P = \left( {2 \times 14 + \frac{{2 \times 10}}{{\sin \,{{90}^{\rm{o}}}}}} \right)\)
\( \Rightarrow P = \left( {28 + \frac{{20}}{1}} \right)\)
\( \Rightarrow P = \left( {28 + 20} \right)\)
\( \Rightarrow P = 48\,{\rm{yards}}\)
Therefore, \(48\,{\rm{yards}}\) is the perimeter of a parallelogram.

Summary

Perimeter is the length of the continuous line formed by its boundary, and its unit is as that of its sides. A parallelogram is developed when the opposite sides of a quadrilateral are parallel and are of equal length. Some of the examples of Parallelograms are squares, rectangles, and rhombus. One can calculate the perimeter of a parallelogram by adding the lengths of all four sides. The parallelogram’s important feature is that it transfers mechanical motion from one location to another.

FAQs on Perimeter of a Parallelogram Formula

Q.1. What is the perimeter of a parallelogram?
Ans: The perimeter is calculated by adding the lengths of all four sides. A parallelogram’s perimeter is the whole distance outside of the geometrical shape. A parallelogram’s opposite sides are equal.

Q.2. What is the formula of the perimeter?
Ans: If a and b are the lengths of the adjacent sides of the parallelogram, then its perimeter is given by:
\(P = 2\left( {a + b} \right)\,{\rm{units}}\)

Q.3. Why is perimeter important?
Ans: The perimeter of a parallelogram has its applications in surveying, flooring estimates, architectural, mechanical engineering, and many more. Knowledge of perimeter plays a major role in everyday life activities.

Q.4. What does the perimeter tell you?
Ans: By the perimeter of a closed figure, we mean the total length of its boundary, obtained by adding the lengths of its sides, either straight or curved or both.

Q.5. What is parallelogram examples?
Ans: A parallelogram is a flat shape with opposite sides parallel and equal. The adjacent pair of angles in a parallelogram is supplementary. Parallelograms include squares, rectangles, and rhombuses.

Q.6. What is the importance of parallelogram in my daily life?
Ans: There are four straight sides to a parallelogram. Each of the two opposing pairs of sides is of the same length, and they are parallel to each other. The parallelogram’s specific features have been used extensively to transfer mechanical motion from one location to another.

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