What Formula Can Be Used to Find the Area of the Parallelogram?
Area of Parallelogram
The area of a parallelogram is defined every bit the region or space covered past a parallelogram in a two-dimensional plane. A parallelogram is a special kind of quadrilateral. If a quadrilateral has ii pairs of parallel opposite sides, so it is called a parallelogram. Rectangle, square, and rhombus are all examples of a parallelogram. Geometry is all near shapes, 2d or 3D. All of these shapes have a different set of backdrop with different formulas for area. The prime focus here volition be entirely on the following:
- Definition of the area of a parallelogram
- Formula of the area of a parallelogram
- Calculation of a parallelogram's expanse in vector class
| 1. | What Is the Surface area of Parallelogram? |
| 2. | Surface area of Parallelogram Formula |
| three. | How To Summate the Expanse of Parallelogram? |
| 4. | Area of Parallelogram in Vector Form |
| 5. | FAQs on Area of Parallelogram |
What Is the Area of Parallelogram?
The expanse of a parallelogram refers to the total number of unit squares that can fit into information technology and it is measured in square units (like cm2, m2, in2, etc). It is the region enclosed or encompassed past a parallelogram in two-dimensional space. Let us remember the definition of a parallelogram. A parallelogram is a four-sided, 2-dimensional effigy with:
- two equal, opposite sides,
- 2 intersecting and non-equal diagonals, and
- opposite angles that are equal
We come across many geometric shapes other than rectangles and squares in our daily lives. Since few backdrop of a rectangle and the parallelogram are somewhat similar, the area of the rectangle is similar to the surface area of a parallelogram.
Surface area of a Parallelogram Formula
The area of a parallelogram tin be calculated by multiplying its base with the altitude. The base and altitude of a parallelogram are perpendicular to each other as shown in the following figure. The formula to calculate the surface area of a parallelogram can thus be given as,
Area of parallelogram = b × h square units
where,
- b is the length of the base
- h is the meridian or distance
Let us analyze the above formula using an case. Assume that PQRS is a parallelogram. Using filigree paper, let us detect its area by counting the squares.
From the above effigy:
Total number of complete squares = 16
Total number of half squares = 8
Area = 16 + (1/2) × viii = 16 + four = xx unit of measurement2
Also, we observe in the figure that ST ⊥ PQ. Past counting the squares, we get:
Side, PQ = 5 units
Corresponding superlative, ST=4 units
Side × tiptop = 5 × four = 20 unit2
Thus, the area of the given parallelogram is base times the altitude.
Permit's do an activity to sympathise the area of a parallelogram.
- Step I: Describe a parallelogram (PQRS) with altitude (SE) on a cardboard and cutting it.
- Pace Ii: Cutting the triangular portion (PSE).
- Step 3: Paste the remaining portion (EQRS) on a white chart.
- Step Iv: Paste the triangular portion (PSE) on the white nautical chart joining sides RQ and SP.
Afterward doing this activity, we observed that the surface area of a rectangle is equal to the area of a parallelogram. Also, the base of operations and height of the parallelogram are equal to the length and breadth of the rectangle respectively.
Area of Parallelogram = Base × Meridian
How To Calculate Area of a Parallelogram?
The parallelogram surface area can be calculated with the assistance of its base and height. Also, the area of a parallelogram can as well be evaluated if its ii diagonals along with any of their intersecting angles are known, or if the length of the parallel sides along with any of the angles betwixt the sides is known.
Parallelogram Area Using Height
Suppose 'a' and 'b' are the ready of parallel sides of a parallelogram and 'h' is the superlative (which is the perpendicular distance betwixt 'a' and 'b'), so the expanse of a parallelogram is given by:
Area = Base × Height
A = b × h [square units]
Case: If the base of a parallelogram is equal to 5 cm and the tiptop is 4 cm, so find its area.
Solution: Given, length of base = 5 cm and pinnacle = 4 cm
As per the formula, Area = five × iv = twenty cm2
Parallelogram Expanse Using Lengths of Sides
The area of a parallelogram can too be calculated without the height if the length of adjacent sides and bending between them are known to us. We can just use the area of the triangle formula from the trigonometry concept for this case.
Area = ab sin (θ)
where,
- a and b = length of parallel sides, and,
- θ = angle betwixt the sides of the parallelogram.
Example: The angle between any two sides of a parallelogram is 90 degrees. If the length of the two parallel sides is 4 units and 6 units respectively, then notice the expanse.
Solution:
Allow a = 4 units and b = 6 units
θ = 90 degrees
Using area of parallelogram formula,
Area = ab sin (θ)
⇒ A = 4 × vi sin (90º)
⇒ A = 24 sin 90º
⇒ A = 24 × ane = 24 sq.units.
Annotation: If the angle between the sides of a parallelogram is ninety degrees, then the parallelogram becomes a rectangle.
Parallelogram Area Using Diagonals
The area of any given parallelogram can likewise be calculated using the length of its diagonals. In that location are 2 diagonals for a parallelogram, intersecting each other at sure angles. Suppose, this angle is given past x, so the surface area of the parallelogram is given past:
Area = ½ × d\(_1\) × d\(_2\) sin (x)
where,
- d\(_1\) and d\(_2\) = Length of diagonals of the parallelogram, and
- x = Angle between the diagonals.
Surface area of Parallelogram in Vector Form
The area of the parallelogram can be calculated using dissimilar formulas even when either the sides or the diagonals are given in the vector form. Consider a parallelogram ABCD as shown in the effigy below,
Area of parallelogram in vector grade using the adjacent sides is,
\(|\overrightarrow{\mathrm{a}} × \overrightarrow{\mathrm{b}}|\)
where, \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are vectors representing 2 adjacent sides.
Hither,
\(\overrightarrow{a} + \overrightarrow{b} = \overrightarrow{d_1} \) → i) and,
\(\overrightarrow{b} + (-\overrightarrow{a}) = \overrightarrow{d_2} \)
or, \(\overrightarrow{b} - \overrightarrow{a} = \overrightarrow{d_2}\) → ii)
⇒ \( \overrightarrow{d_1} \times \overrightarrow{d_2} = (\overrightarrow{a} + \overrightarrow{b}) (\overrightarrow{b} - \overrightarrow{a})\)
= \(\overrightarrow{a}\) × (\(\overrightarrow{b}\) - \(\overrightarrow{a}\)) + \(\overrightarrow{b}\) × (\(\overrightarrow{b}\) - \(\overrightarrow{a}\))
= \(\overrightarrow{a}\) × \(\overrightarrow{b}\) - \(\overrightarrow{a}\) × \(\overrightarrow{a}\) + \(\overrightarrow{b}\) × \(\overrightarrow{b}\) - \(\overrightarrow{b}\) × \(\overrightarrow{a}\)
Since \(\overrightarrow{a}\) × \(\overrightarrow{a}\) = 0, and \(\overrightarrow{b}\) × \(\overrightarrow{b}\) = 0
⇒ \(\overrightarrow{a}\) × \(\overrightarrow{b}\) - 0 + 0 - \(\overrightarrow{b}\) × \(\overrightarrow{a}\)
Since \(\overrightarrow{a}\) × \(\overrightarrow{b}\) = - \(\overrightarrow{b}\) × \(\overrightarrow{a}\),
\( \overrightarrow{d_1}\) × \(\overrightarrow{d_2}\) = \(\overrightarrow{a}\) × \(\overrightarrow{b}\) - (-(\(\overrightarrow{a}\) × \(\overrightarrow{b}\)))
= 2(\(\overrightarrow{a}\) × \(\overrightarrow{b}\))
Therefore, expanse of parallelogram when diagonals are given in vector form = one/ii |(\(\overrightarrow{d_1}\) × \(\overrightarrow{d_2}\))|
where, \(\overrightarrow{d_1}\) and \(\overrightarrow{d_2}\) are diagonals.
Thinking Out Of the Box!
- Tin a kite be called a parallelogram?
- What elements of a trapezoid should be inverse to make it a parallelogram?
- Can there be a concave parallelogram?
- Tin you detect the area of the parallelogram without knowing its height?
Area of Parallelogram Examples
-
Case 1: The side by side sides of a parallelogram are 10 in and 6 in. The altitude corresponding to side x in is five in. Using the area of parallelogram formula, find the area and the length of the altitude corresponding to its next side.
Solution:
Let ABCD be a parallelogram where DE⊥AB, AF⊥BC
Using the area of parallelogram formula,
Surface area of Parallelogram ABCD = (10) × (5) = 50 in2
Length of BC = 6 in
Length of Distance AF = (l ÷ half-dozen) = 8.3 inRespond: Surface area of the given parallelogram = 50 in2; Distance = 8.3 in
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FAQs on Surface area of Parallelogram
What Is the Area of a Parallelogram in Math?
The expanse of a parallelogram is defined as the region enclosed or encompassed by a parallelogram in ii-dimensional space. It is represented in square units like cmii, one thousand2, in2, etc.
How To Notice Area of a Parallelogram Without Height?
The area of a parallelogram can be calculated without the peak when the length of adjacent sides and the angle between them is known. The formula to find the area for this case is given as, area = ab sin (θ), where 'a' and 'b' are lengths of adjacent sides, and, θ is the angle between them.
Also, the surface area tin exist calculated when the diagonals and their intersecting bending are given, using the formula, Area = ½ × d1 × d2 sin (y), where 'd1' and 'd2' are lengths of diagonals of the parallelogram, and 'y' is the angle between them.
What Is the Formula of Finding Expanse of Parallelogram?
The area of a parallelogram can exist calculated past finding the product of its base of operations with the distance. The base and altitude of a parallelogram are always perpendicular to each other. The formula to summate the expanse of a parallelogram is given equally Surface area of parallelogram = base × top square units.
How To Find Area of Parallelogram With Vectors?
Expanse of a parallelogram tin can be calculated when the adjacent sides or diagonals are given in the vector form. The formula to notice area using vector next sides is given every bit, | \(\overrightarrow{a}\) × \(\overrightarrow{b}\)|, where \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are side by side side vectors. Also, the area of parallelogram formula using diagonals in vector grade is, area = one/two |(\(\overrightarrow{d_1}\) × \(\overrightarrow{d_2}\))|, where \(\overrightarrow{d_1}\) and \(\overrightarrow{d_2}\) are diagonal vectors.
How to Calculate Area of Parallelogram Using Reckoner?
To make up one's mind the surface area of a parallelogram the easiest and fastest method is to use the area of a parallelogram figurer. Information technology is a free online tool that helps you to summate the area of a parallelogram with the assistance of the given dimensions. Try now Cuemath'due south area of parallelogram calculator, enter the value of height and base of the parallelogram and get the parallelogram's area within a few seconds.
What Is the Area of a Parallelogram When Diagonals are Given?
The area of a parallelogram can be calculated when the diagonals and their intersecting bending are known. The formula is given equally, expanse = ½ × d1 × d2 sin (x), where 'd1' and 'd2' are lengths of diagonals of the parallelogram, and 'x' is the angle betwixt them.
How To Summate Area of Parallelogram Whose Adjacent Sides are Given?
To detect the area of a parallelogram when the lengths of next sides are given, we need the bending between them. The formula to find the expanse for this case is given as, expanse = ab sin (θ), where 'a' and 'b' are lengths of adjacent sides, and, θ is the angle betwixt the sides of the parallelogram.
Source: https://www.cuemath.com/measurement/area-of-parallelogram/
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