Parallelogram: Definition, Properties, Formulas, Types

A parallelogram is a geometrical shape with sides parallel to each other in two dimensions. It’s a four-sided polygon (sometimes known as a quadrilateral) with two parallel sides of the same length. A parallelogram’s sum of neighboring angles equals 180 degrees. 

Building structures, rangoli designs, tiles, tables, etc., are the few things we can commonly see parallelogram shape. The name “parallelogram” comes from the Greek word “parallelogrammon,” which means “bounded by parallel lines.” Let us learn more about What is a Parallelogram, its properties with examples.

What is a Parallelogram?

Parallelogram is a quadrilateral (four-sided plane figure) whose opposite sides are equal and parallel. The Sum of interior angles of the parallelogram is \(360°\) and the sum of the adjacent angles of a parallelogram is \(180°\).

Properties of a Parallelogram

Properties of a parallelogram are:

1. The opposite sides of the parallelogram are parallel.
2. The opposite sides of the parallelogram are equal.
3. The opposite angles of the parallelogram are equal.
4. The diagonals of the parallelogram bisect each other.

The converse of the parallelogram properties are also true i.e.,

1. The quadrilateral is a parallelogram if it has one pair of opposite sides parallel and equal.
2. The quadrilateral is a parallelogram if its opposite sides are equal.
3. The quadrilateral is a parallelogram if its opposite angles are equal.
4. The quadrilateral is a parallelogram if its diagonals bisect each other.

Area of Plane Figure

We have seen that the part of the plane enclosed by a simple closed figure is called the region enclosed by the figure and the measurement of the enclosed region is called its area. Thus, by the area of a plain simple closed figure, we mean the measurement of the region enclosed by the figure.

Parallelogram Formula:

Area of a parallelogram

The area of a parallelogram is the region covered inside the boundary (sides) of a parallelogram. The area of the Parallelogram is the product of its base and height.

\( {\text{Area}}\, {\text{of}}\, {\text{a}}\, {\text{parallelogram}}={\text{base}} \times {\text{height}}\)

If length of base of a parallelogram is \(b\,\text{cm}\) and corresponding altitude is \(h\,\text{cm}\), then.
\({\text{Area}}\,(A)\, {\text{of}}\, {\text{parallelogram}} = b \times h\, {\text{c}} { {\text{m}}^2}\)

The perimeter of a parallelogram: Perimeter of a parallelogram is the total length of the sides of a parallelogram.

Since opposite sides of a parallelogram are equal, the perimeter of a parallelogram is twice the sum of adjacent sides.

That is \( {\text{Perimeter}} = 2({\text{sum}}\, {\text{of}}\, {\text{adjacent}}\, {\text{sides}})\)

Unit of Area

Unit of area are \( {\text{c}} { {\text{m}}^ {\text{2}}}, {\text{m}}^ {\text{2}}, {\text{k}} { {\text{m}}^ {\text{2}}}\), etc. \( {\text{1}}\, {\text{c}} { {\text{m}}^ {\text{2}}}\) is the area of the region enclosed by a square of side \(1\,\text{cm}\).

Other standard units of area and inter-relationship between them:

\( {\text{100}}\, {\text{m}} { {\text{m}}^ {\text{2}}} {\text{=1}}\, {\text{c}} { {\text{m}}^ {\text{2}}} {\text{,}}\, {\text{100}}\, { {\text{m}}^ {\text{2}}} {\text{=1}}\, {\text{Are}}\, {\text{(da}} { {\text{m}}^ {\text{2}}} {\text{)}}\)

\( {\text{100}}\, {\text{c}} { {\text{m}}^ {\text{2}}} {\text{=1}}\, {\text{d}} { {\text{m}}^ {\text{2}}} {\text{,}}\, {\text{100}}\, {\text{Are=1}}\, {\text{hectare/}} {\left({{\text{hectometre}}} \right)^2}\)

\( {\text{100}}\, {\text{d}} { {\text{m}}^ {\text{2}}} {\text{=1}}\, { {\text{m}}^ {\text{2}}} {\text{,}}\, {\text{100}}\, {\text{hectare=1}}\, {\text{k}} { {\text{m}}^ {\text{2}}}\)

\( {\text{10000}}\, {\text{c}} { {\text{m}}^ {\text{2}}} {\text{=1}}\, { {\text{m}}^ {\text{2}}}\)

Note: Are, unit of area in the metric system, equal to \( {\text{100}}\, { {\text{m}}^ {\text{2}}}\) and the equivalent of \(0.0247\, {\text{acre}}\).

Parallelogram Shape

A quadrilateral is a parallelogram if it’s both the pairs of opposite sides are parallel.

Special Types of Parallelogram

1. Square
2. Rhombus
3. Rectangle

Square: A Parallelogram having all sides equal and each angle measure equal to the right angle is called a Square.

Properties of a Square

1. All the sides are of the same length.
2. Each of the angles is a right angle.
3. The diagonals are of equal length.
4. The diagonals bisect each other at right angles.

Rhombus: A parallelogram having all sides equal is called a rhombus and opposite angles are equal.

Properties of Rhombus

1. All the sides of a rhombus are equal.
2. The opposite angles of a rhombus are equal.
3. The adjacent angles of a rhombus are supplementary.
4. The diagonals of a rhombus bisect each other at \({90^\circ }\).

Rectangle: A parallelogram whose each angle is \({90^\circ }\) is called a rectangle.

Properties of a Rectangle

• Each angle of a rectangle is \({90^\circ }\).
• The diagonals of a rectangle are equal in measure.

Note: Since a rhombus, a rectangle and a square are special types of a parallelogram, so all the properties of a parallelogram are true for each of them.

Angles of Parallelogram

1. In a parallelogram the sum of any two adjacent angles (the angles having a common) arm are supplementary.
Here in parallelogram \(ABCD\), \(\angle A + \angle D = {180^ \circ }\), \(\angle A + \angle B = {180^ \circ }\), \(\angle B + \angle C = {180^ \circ }\) and \(\angle C + \angle D = {180^ \circ }\)
2. The opposite angles of a parallelogram are of equal in measure.
Here in parallelogram \(ABCD\), \(\angle A = \angle C\) and \(\angle B = \angle D\).
3. The sum of the interior angles of a parallelogram is \(360°\).
Here in parallelogram \(ABCD\), \(\angle A + \angle B + \angle C + \angle D = {360^ \circ }\).

Solved Examples

Q.1. Find the area of a parallelogram, whose base is \({\rm{10\,{cm}}}\) and height is \({\rm{5\,{cm}}}\). Ans: Given:
Base \((b)\) of a parallelogram =\(10\,\rm{cm}\)
Height \((h)\) of a parallelogram \(5\,\rm{cm}\)
We know that area of a parallelogram \( = b \times h\)
Area \((A)\) of a parallelogram \( = 10 \times 5 = 50\,\rm{c {m^2}}\)
Hence, the area of the given parallelogram is \(50\,\rm{c {m^2}}\).

Q.2. Area of a parallelogram shaped plot is \(7500\, { {\text{m}}^ {\text{2}}}\) and its altitude is \({75\,\rm{m}}\). Find the base of a parallelogram shaped plot.
Ans: Given: Area of a parallelogram-shaped plot \(7500\, { {\text{m}}^ {\text{2}}}\)
Altitude \((h)\) of a parallelogram shaped plot \(75\,\rm{m}\)
We know that area of a parallelogram \( = b \times h\)
\(7500\,\rm{m^2} = b \times 75\,\rm{m}\)
\( \Rightarrow b = \frac{{7500}}{{75}} = 100\,{\text{m}}\)
Hence, the base of a given parallelogram shaped plot is \(100\,\rm{m}\)

Q.3. Area of a parallelogram shaped park is \({\rm{6250\;}}{{\rm{m}}^{\rm{2}}}\) and its base is \({250\,\rm{m}}\). Find the altitude of a parallelogram shaped park.
Ans: Given:
Area of a parallelogram-shaped park \({\rm{6250\;}}{{\rm{m}}^{\rm{2}}}\)
Base \((b)\) of a parallelogram shaped park \(= 250\,\rm{m}\)
We know that area of a parallelogram \( = b \times h\)
\(6250\,{{\text{m}}^2} = 250\,{\text{m}} \times {\text{h}}\)
\( \Rightarrow h = \frac{ {6250}}{ {250}} = 25\,\rm{m}\)
Hence, the altitude of a given parallelogram shaped park is \(25\,\rm{m}\).

Q.4. Find the perimeter of a parallelogram, whose adjacent sides are \({15\,\rm{cm}}\) and \({8\,\rm{cm}}\).
Ans: Given:
Let the adjacent sides of the given parallelogram be \(a\) and \(b\).
So, \(a = 15\,\rm{cm}\) and \(b = 8\,\rm{cm}\)
We know that perimeter of a parallelogram \( = 2\left(\rm{sum}\,\rm{of}\,\rm{adjacent}\,\rm{sides} \right)\)
Now, the perimeter of a given parallelogram \( = 2\left({a + b} \right) = 2\left({15 + 8} \right)\)
\( = 2 \times 23 = 46\,\rm{cm}\)
Hence, the perimeter of a given parallelogram is \(46\,\rm{cm}\).

Q.5. The perimeter of a parallelogram-shaped plot is \(400\,\rm{m}\) and one of its adjacent side is \(150\,\rm{m}\). Find the measure of its another adjacent side.
Ans: Given:
The perimeter of a parallelogram shaped plot \(= 400\,\rm{m}\)
One of the adjacent side of a parallelogram shaped plot \(= 150\,\rm{m}\)
Let another adjacent side of a parallelogram shaped plot be \(x\,\rm{m}\)
We know that perimeter of a parallelogram \( = 2\left({{\text{sum}}\,{\text{of}}\, {\text{adjacent}}\, {\text{sides}}} \right)\)
Now, \(400\, {\text{m}} = 2(150 + x)\,\rm{m}\)
\( \Rightarrow 200\,{\rm{m}}\,{\rm{ = }}\,\left( {150 + x} \right)\,{\rm{m}}\)
\( \Rightarrow x = 50\,\rm{m}\).
Hence, another adjacent side of a given parallelogram is \(50\,\rm{m}\).

Summary

The learning outcome from the parallelogram topic will help in understanding all the geometrical shapes related to a parallelogram, how to calculate the area and perimeter of a parallelogram and where we can see the parallelogram shape in our daily life.

FAQs

Q.1. What shape is a parallelogram?
Ans: A parallelogram is a two-dimensional figure, having pairs of parallel sides. The opposite sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

Q.2: What are the special types of parallelograms?
Ans: Square, rhombus and rectangle are few special types of a parallelogram.
Since a rhombus, a rectangle and a square are special types of a parallelogram, so all the properties of a parallelogram are true for each of them.

Q.3: What are the properties of a parallelogram?
Ans: Properties of a parallelogram:
1. The opposite sides of a parallelogram are parallel.
2. The opposite sides of a parallelogram are equal.
3. The opposite angles of a parallelogram are equal.
4. The diagonals of a parallelogram bisect each other.
5. In a parallelogram the sum of any two adjacent angles (The angles having a common) arm is supplementary. 

Q.4. What is the formula to find the area of a parallelogram?
Ans: The formula to find the area of a parallelogram is \(b\times h\).
Where \(b\) is the base of a parallelogram and \(h\) is the height (altitude) of a parallelogram.

Q.5. What is the formula to find the perimeter of a parallelogram?
Ans: Formula to find the perimeter of a parallelogram is \( 2\left({{\text{sum}}\, {\text{of}}\, {\text{adjacent}}\, {\text{sides}}} \right)\)

Q.6. What is the unit of area of a parallelogram?
Ans: Unit of an area of a parallelogram is square centimetre, square meter, etc.
It depends on the units for the measures of base and height of a parallelogram given.

Q.7. Are all the rhombuses parallelograms?
Ans: A parallelogram having all sides equal and opposite angles are equal is called a rhombus.
Therefore, all the rhombuses are parallelograms, but all the parallelograms are not rhombuses. 

Q.8. What are real-life examples of a parallelogram?
Ans: Building structures, rangoli designs, tiles, tables, etc. are the few things in which we can commonly see parallelogram shape.

Q.9: How do you prove a parallelogram?
Ans:
1. The quadrilateral is a parallelogram if it has one pair of opposite sides parallel and equal.
2. The quadrilateral is a parallelogram if its opposite sides are equal.
3. The quadrilateral is a parallelogram if its opposite angles are equal.
4. The quadrilateral is a parallelogram if its diagonals bisect each other.

Q.10. Are all the rectangles parallelograms?
Ans: A parallelogram whose each angle is \({90^ \circ }\) called a rectangle.
Therefore, all the rectangles are parallelograms, but all the parallelograms are not rectangles.

Q.11. Are all the parallelograms squares?
Ans: A parallelogram having all sides equal and each angle measure equal to a right angle, is called a square.
Therefore, all the parallelograms are not square, but all the squares are parallelograms.

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