✅ Parallelogram Formula ⭐️⭐️⭐️⭐️⭐️

Area of Parallelogram

Area of a parallelogram is a region covered by a parallelogram in a two-dimensional plane. In Geometry, a parallelogram is a two-dimensional figure with four sides. It is a special case of the quadrilateral, where opposite sides are equal and parallel. The area of a parallelogram is the space enclosed within its four sides. Area is equal to the product of length and height of the parallelogram.

The sum of the interior angles in a quadrilateral is 360 degrees. A parallelogram has two pairs of parallel sides with equal measures. Since it is a two-dimensional figure, it has an area and perimeter. In this article, let us discuss the area of a parallelogram with its formula, derivations, and more solved problems in detail.

What is the Area of Parallelogram?

The area of a parallelogram is the region bounded by the parallelogram in a given two-dimension space. To recall, a parallelogram is a special type of quadrilateral which has four sides and the pair of opposite sides are parallel. In a parallelogram, the opposite sides are of equal length and opposite angles are of equal measures. Since the rectangle and the parallelogram have similar properties, the area of the rectangle is equal to the area of a parallelogram.

Area of Parallelogram Formula

To find the area of the parallelogram, multiply the base of the perpendicular by its height. It should be noted that the base and the height of the parallelogram are perpendicular to each other, whereas the lateral side of the parallelogram is not perpendicular to the base. Thus, a dotted line is drawn to represent the height.

Therefore,

Area = b × h Square units

Where “b” is the base and “h” is the height of the parallelogram.

Let us learn the derivation of area of a parallelogram, in the next section.

How to Calculate the Area of Parallelogram?

The parallelogram area can be calculated, using its base and height. Apart from it, the area of a parallelogram can also be evaluated, if its two diagonals are known along with any of their intersecting angles, or if the length of the parallel sides is known, along with any of the angles between the sides. Hence, there are three method to derive the area of parallelogram:

• When base and height of parallelogram are given
• When height is not given
• When diagonals are given

Area of Parallelogram Using Sides

Suppose a and b are the set of parallel sides of a parallelogram and h is the height, then based on the length of sides and height of it, the formula for its area is given by:

Area = Base × Height

A = b × h     [sq.unit]

Example: If the base of a parallelogram is equal to 5 cm and the height is 3 cm, then find its area.

Solution: Given, length of base=5 cm and height = 3 cm

As per the formula, Area = 5 × 3 = 15 sq.cm

Area of Parallelogram Without Height

If the height of the parallelogram is unknown to us, then we can use the trigonometry concept here to find its area.

Area = ab sin (x)

Where a and b are the length of parallel sides and x is the angle between 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 3 cm and 4 cm respectively, then find the area.

Solution: Let a = 3 cm and b=4 cm

x = 90 degrees

Area = ab sin (x)

A = 3 × 4 sin (90)

A = 12 sin 90

A = 12 × 1 = 12 sq.cm.

Note: If the angle between the sides of a parallelogram is 90 degrees, then it is a rectangle.

Area of Parallelogram Using Diagonals

The area of any parallelogram can also be calculated using its diagonal lengths. As we know, there are two diagonals for a parallelogram, which intersects each other. Suppose, the diagonals intersect each other at an angle y, then the area of the parallelogram is given by:

Area = ½ × d₁ × d₂ sin (y)

Check the table below to get summarised formulas of an area of a parallelogram.

All Formulas to Calculate Area of a Parallelogram
Using Base and Height	A = b × h
Using Trigonometry	A = ab sin (x)
Using Diagonals	A = ½ × d1 × d2 sin (y)

Where,

• b = base of the parallelogram (AB)
• h = height of the parallelogram
• a = side of the parallelogram (AD)
• x = any angle between the sides of the parallelogram (∠DAB or ∠ADC)
• d₁ = diagonal of the parallelogram (p)
• d₂ = diagonal of the parallelogram (q)
• y = any angle between at the intersection point of the diagonals (∠DOA or ∠DOC)

Note: In the above figure,

• DC = AB = b
• AD = BC = a
• ∠DAB = ∠DCB
• ∠ADC = ∠ABC
• O is the intersecting point of the diagonals
• ∠DOA = ∠COB
• ∠DOC = ∠AOB

Area of Parallelogram in Vector Form

If the sides of a parallelogram are given in vector form then the area of the parallelogram can be calculated using its diagonals. Suppose, vector ‘a’ and vector ‘b’ are the two sides of a parallelogram, such that the resulting vector is the diagonal of parallelogram.

Area of parallelogram in vector form = Mod of cross-product of vector a and vector b

A = | a × b|

Now, we have to find the area of a parallelogram with respect to diagonals, say d₁ and d₂,  in vector form.

So, we can write;

a + b = d₁

b + (-a) = d₂

or

b – a = d₂

Thus,

d₁ × d₂ = (a + b) × (b – a)

= a × (b – a) + b × (b – a)

= a × b – a × a + b × b – b × a

= a × b – 0 + 0 – b × a

= a × b – b × a

Since,

a × b = – b × a

Therefore,

d₁ × d₂ = a × b + a × b = 2 (a × b)

a × b = 1/2 (d₁ × d₂)

Hence,

Area of parallelogram when diagonals are given in the vector form, becomes:

A = 1/2 (d₁ × d₂)

where d1 and d2 are vectors of diagonals.

Example: Find the area of parallelogram whose adjacent sides are given in vectors.

A = 3i + 2j and B = -3i + 1j

Area of parallelogram = |A × B|

= i (0-0) – (0-0) + k(3+6)  [Using determinant of 3 x 3 matrix formula]

= 9k

Thus, area of the parallelogram formed by two vectors A and B is equal to 9k sq.unit.

Solved Examples on Area of Parallelogram

Question 1: Find the area of the parallelogram with the base of 4 cm and height of 5 cm.

Solution:

Given:

Base, b = 4 cm

h = 5 cm

We know that,

Area of Parallelogram = b×h Square units

= 4 × 5 = 20 sq.cm

Therefore, the area of a parallelogram = 20 cm²

Question 2: Find the area of a parallelogram whose breadth is 8 cm and height is 11 cm.

Solution:

Given,

b = 8 cm

h = 11 cm

Area of a parallelogram

= b × h

= 8 × 11 cm²

= 88 cm²

Question 3: The base of the parallelogram is thrice its height. If the area is 192 cm², find the base and height.

Solution:

Let the height of the parallelogram = h cm

then, the base of the parallelogram = 3h cm

Area of the parallelogram = 192 cm²

Area of parallelogram = base × height

Therefore, 192 = 3h × h

⇒ 3 × h² = 192

⇒ h² = 64

⇒ h = 8 cm

Hence, the height of the parallelogram is 8 cm, and breadth is

3 × h

= 3 × 8

= 24 cm

Word Problem on Area of Parallelogram

Question: The area of a parallelogram is 500 sq.cm. Its height is twice its base. Find the height and base.

Solution:

Given, area = 500 sq.cm.

Height = Twice of base

h = 2b

By the formula, we know,

Area = b x h

500 = b x 2b

2b² = 500

b² = 250

b = 15.8 cm

Hence, height = 2 x b = 31.6 cm

Practise Questions on Area of a Parallelogram

1. Find the area of a parallelogram whose base is 8 cm and height is 4 cm.
2. Find the area of a parallelogram with a base equal to 7 inches and height is 9 inches.
3. The base of the parallelogram is thrice its height. If the area is 147 sq.units, then what is the value of its base and height?
4. A parallelogram has sides equal to 10m and 8m. If the distance between the shortest sides is 5m, then find the distance between the longest sides of the parallelogram. (Hint: First find the area of parallelogram using distance between shortest sides)

Frequently Asked Questions

What is a Parallelogram?

A parallelogram is a geometrical figure that has four sides formed by two pairs of parallel lines. In a parallelogram, the opposite sides are equal in length, and opposite angles are equal in measure.

What is the Area of a Parallelogram?

The area of any parallelogram can be calculated using the following formula:

Area = base × height

It should be noted that the base and height of a parallelogram must be perpendicular.

What is the Perimeter of a Parallelogram?

To find the perimeter of a parallelogram, add all the sides together. The following formula gives the perimeter of any parallelogram:

Perimeter = 2 (a + b)

What is the Area of a Parallelogram whose height is 5 cm and base is 4 cm?

The area of a perpendicular with height 5 cm and base 4 cm will be;

A = b × h

Or, A = 4 × 5 = 20 cm²

Parallelogram

A parallelogram is a two-dimensional geometrical shape, whose sides are parallel to each other. It is a type of polygon having four sides (also called quadrilateral), where the pair of parallel sides are equal in length. The Sum of adjacent angles of a parallelogram is equal to 180 degrees. In geometry, you must have learned about many 2D shapes and sizes such as circle, square, rectangle, rhombus, etc. All of these shapes have a different set of properties. Also, the area and perimeter formulas of these shapes vary from each other and are used to solve many problems. Let us learn here the definition, formulas and properties of a parallelogram.

Parallelogram Definition

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure. Also, the interior angles on the same side of the transversal are supplementary. Sum of all the interior angles equals 360 degrees.

A three-dimensional shape that has its faces in parallelogram shape, is called a parallelepiped. The area of parallelogram depends on the base (one of its parallel sides) and height (altitude drawn from top to bottom) of it. The perimeter of a parallelogram depends on the length of its four sides.

A square and a rectangle are two shapes which have similar properties of a parallelogram.

Rhombus: If all the sides of a parallelogram are congruent or equal to each other, then it is a rhombus.

If there is one parallel side and the other two sides are non-parallel, then it is a trapezium.

See the figure below:

In the figure above, you can see, ABCD is a parallelogram, where AB || CD and AD || BC. 

Also, AB = CD and AD = BC

And, ∠A = ∠C & ∠B = ∠D

Also, ∠A & ∠D are supplementary angles because these interior angles lie on the same side of the transversal. In the same way, ∠B & ∠C are supplementary angles.

Therefore,

∠A + ∠D = 180

∠B + ∠C = 180

Facts:

• Number of sides = 4
• Number of vertices = 4
• Mutually Parallel sides = 2 (in pair)
• Area = Base x Height
• Perimeter = 2 (Sum of adjacent sides length)
• Type of polygon = Quadrilateral

Shape of Parellelogram

A parallelogram is a two-dimensional shape. It has four sides, in which two pairs of sides are parallel. Also, the parallel sides are equal in length. If the length of the parallel sides is not equal in measurement, then the shape is not a parallelogram. Similarly, the opposite interior angles of parallelogram should always be equal. Otherwise, it is not a parallelogram.

Special Parallelograms

Square and Rectangle: A square and a rectangle are two shapes which have similar properties of a parallelogram. Both have their opposite sides equal and parallel to each other. Diagonals of both shapes bisect each other. 

Rhombus: If all the sides of a parallelogram are congruent or equal to each other, then it is a rhombus.

Rhomboid: A special case of a parallelogram that has its opposite sides parallel to each other but adjacent sides are of unequal lengths. Also, the angles are equal to 90 degrees.

Trapezium: If there is one parallel side and the other two sides are non-parallel, then it is a trapezium. 

Angles of Parallelogram

A parallelogram is a flat 2d shape which has four angles. The opposite interior angles are equal. The angles on the same side of the transversal are supplementary, that means they add up to 180 degrees. Hence, the sum of the interior angles of a parallelogram is 360 degrees.

Properties of Parallelogram

If a quadrilateral has a pair of parallel opposite sides, then it’s a special polygon called Parallelogram. The properties of a parallelogram are as follows:

• The opposite sides are parallel and congruent
• The opposite angles are congruent
• The consecutive angles are supplementary
• If any one of the angles is a right angle, then all the other angles will be at right angle
• The two diagonals bisect each other
• Each diagonal bisects the parallelogram into two congruent triangles
• The Sum of square of all the sides of parallelogram is equal to the sum of square of its diagonals. It is also called parallelogram law

Formulas (Area & Perimeter)

The formula for the area and perimeter of a parallelogram is covered here in this section. Students can use these formulas and solve problems based on them.

Area of Parallelogram

Area of a parallelogram is the region occupied by it in a two-dimensional plane. Below is the formula to find the parallelogram area:

Area = Base × Height

In the above figure, ||^gramABCD,  Area is given by;

Area = a b sin A = b a sin B

where a is the slant length of the side of ||^gramABCD and b is the base.

Perimeter of Parallelogram

The perimeter of any shape is the total distance covered around the shape or the total length of any shape. Similarly, the perimeter of a parallelogram is the total distance of the boundaries of the parallelogram. To calculate the perimeter value, we have to know the values of its length and breadth. The parallelogram has its opposite sides equal in length. Therefore, the formula to calculate the perimeter is written as;

Perimeter = 2 (a+b) units

Where a and b are the length of the sides of the parallelogram.

Types of Parallelogram

There are mainly four types of Parallelogram, depending on various factors. The factors which distinguish between all of these different types of parallelogram are angles, sides etc.

1. In a parallelogram, say PQRS 
   • If PQ = QR = RS = SP are the equal sides, then it’s a rhombus. All the properties are the same for rhombus as for parallelogram.
2. Other two special types of a parallelogram are:
   • Rectangle
   • Square

Is Square a Parallelogram?

Square could be considered as a parallelogram since the opposite sides are parallel to each other, and the diagonals of the square bisect each other.

Is Rectangle a Parallelogram?

Yes, a rectangle is also a parallelogram, because it satisfies the conditions or meets the properties of parallelogram such as the opposite sides are parallel and diagonals bisect each other.

Parallelogram Theorems

Theorem 1: Parallelograms on the same base and between the same parallel sides are equal in area.

Proof: Two parallelograms ABCD and ABEF, on the same base DC and between the same parallel line AB and FC.

To prove that area (ABCD) = area (ABEF).

Proof:

Consider the figure given below:
Parallelogram ABCD and rectangle ABML are on the same base and between the same parallels AB and LC.

area of parallelogram ABCD = area of parallelogram ABML

We know that area of a rectangle = length x breadth.

Therefore, area of parallelogram ABCD = AB x AL
Hence, the area of a parallelogram is the product of any base of it and the corresponding altitude.

In ∆ADF and ∆BCE,

AD=BC (∴ABCD is a parallelogram ∴ AD=BC)

AF=BE (∴ABEF is a parallelogram ∴AF=BE)

∠ADF=∠BCE (Corresponding Angles)

∠AFD=∠BEC (Corresponding Angles)

∠DAF =∠CBE (Angle Sum Property)

∆ADE ≅ ∆BCF (From SAS-rule)

Area(ADF) = Area(BCE) (By congruence area axiom)

Area(ABCD)=Area(ABED) + Area(BCE)

Area(ABCD)=Area(ABED)+Area(ADF)

Area(ABCD)=Area(ABEF)

Hence, the area of parallelograms on the same base and between the same parallel sides is equal.

Corollary

A parallelogram and a rectangle on the same base and between the same parallels are equal in area.

Proof: Since a rectangle is also a parallelogram so, the result is a direct consequence of the above theorem.

Theorem: The area of a parallelogram is the product of its base and the corresponding altitude.

Given: In a parallelogram ABCD, AB is the base.

To prove that Area(||^gmABCD) = AB×AL

Construction: Complete the rectangle ALMB by Drawing BM perpendicular to CD.

Difference Between Parallelogram and Rhombus

Parallelogram	Rhombus
A quadrilateral that has its opposite sides equal and parallel	A quadrilateral that has all its sides congruent
Diagonals bisect each other	Diagonals bisect each other at 90 degrees
Opposite angles are of equal measure	All four angles are of equal measure

Examples on Parallelogram

The angle opposite to the side b comes out to be 180 – 65 = 115°

We use the law of cosines to calculate the base of the parallelogram –

b² =  5² + 11² – 2(11)(5)cos(115°)

b² =  25 + 121 – 110(-0.422)

b² = 192.48

b = 13.87 cm.

After finding the base, we need to calculate the height of the given parallelogram.

To find the height we have to calculate the value of θ, so we use sine law

5/sin(θ) = b/sin(115)
θ = 19.06

Now we extend the base and draw in the height of the figure and denote it as ‘h’.

The right-angled triangle (marked with a red line) has the Hypotenuse to be 22 cm and the Perpendicular to be h.

So

sin θ = h/22

h = 7.184 cm

Area = base × height

A = 13.87 × 7.184

A = 99.645 sq.cm

Frequently Asked Questions on Parallelogram

What is the shape of a parallelogram?

A parallelogram is a four-sided two-dimensional shape, with two pairs of sides parallel and equal.

What are the four important properties of a parallelogram?

Opposite sides are parallel and congruent
Opposite angles are congruent
Consecutive angles are supplementary
Diagonals bisect each other

What Is the Area of Parallelogram?

The area of a parallelogram refers to the total number of unit squares that can fit into it and it is measured in square units (like cm², m², in², etc). It is the region enclosed or encompassed by a parallelogram in two-dimensional space. Let us recall the definition of a parallelogram. A parallelogram is a four-sided, 2-dimensional figure with:

• two equal, opposite sides,
• two 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 properties of a rectangle and the parallelogram are somewhat similar, the area of the rectangle is similar to the area of a parallelogram.

Area of a Parallelogram Formula

The area of a parallelogram can 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 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 height or altitude

Let us analyze the above formula using an example. Assume that PQRS is a parallelogram. Using grid paper, let us find its area by counting the squares.

From the above figure:
Total number of complete squares = 16
Total number of half squares = 8
Area = 16 + (1/2) × 8 = 16 + 4 = 20 unit²

Also, we observe in the figure that ST ⊥ PQ. By counting the squares, we get:
Side, PQ = 5 units
Corresponding height, ST=4 units
Side × height = 5 × 4 = 20 unit²

Thus, the area of the given parallelogram is base times the altitude.

Let’s do an activity to understand the area of a parallelogram.

• Step I: Draw a parallelogram (PQRS) with altitude (SE) on a cardboard and cut it.
• Step II: Cut the triangular portion (PSE).
• Step III: Paste the remaining portion (EQRS) on a white chart.
• Step IV: Paste the triangular portion (PSE) on the white chart joining sides RQ and SP.

After doing this activity, we observed that the area of a rectangle is equal to the area of a parallelogram. Also, the base and height of the parallelogram are equal to the length and breadth of the rectangle respectively.

Area of Parallelogram = Base × Height

How To Calculate Area of a Parallelogram?

The parallelogram area can be calculated with the help of its base and height. Also, the area of a parallelogram can also be evaluated if its two diagonals along with any of their intersecting angles are known, or if the length of the parallel sides along with any of the angles between the sides is known.

Parallelogram Area Using Height

Suppose ‘a’ and ‘b’ are the set of parallel sides of a parallelogram and ‘h’ is the height (which is the perpendicular distance between ‘a’ and ‘b’), then the area of a parallelogram is given by:

Area = Base × Height

A = b × h [square units]

Example: If the base of a parallelogram is equal to 5 cm and the height is 4 cm, then find its area.

Solution: Given, length of base = 5 cm and height = 4 cm

As per the formula, Area = 5 × 4 = 20 cm²

Parallelogram Area Using Lengths of Sides

The area of a parallelogram can also be calculated without the height if the length of adjacent sides and angle between them are known to us. We can simply 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 between 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 find the area.

Solution:

Let a = 4 units and b = 6 units
θ = 90 degrees

Using area of parallelogram formula,
Area = ab sin (θ)
⇒ A = 4 × 6 sin (90º)
⇒ A = 24 sin 90º
⇒ A = 24 × 1 = 24 sq.units.

Note: If the angle between the sides of a parallelogram is 90 degrees, then the parallelogram becomes a rectangle.

Parallelogram Area Using Diagonals

The area of any given parallelogram can also be calculated using the length of its diagonals. There are two diagonals for a parallelogram, intersecting each other at certain angles. Suppose, this angle is given by x, then the area of the parallelogram is given by:

Area = ½ × d1 × d2

sin (x)

where,

• d1

and d2 = Length of diagonals of the parallelogram, and x = Angle between the diagonals.

Area of Parallelogram in Vector Form

The area of the parallelogram can be calculated using different formulas even when either the sides or the diagonals are given in the vector form. Consider a parallelogram ABCD as shown in the figure below,

Area of parallelogram in vector form using the adjacent sides is,

Thinking Out Of the Box!

• Can a kite be called a parallelogram?
• What elements of a trapezoid should be changed to make it a parallelogram?
• Can there be a concave parallelogram?
• Can you find the area of the parallelogram without knowing its height?

Area of Parallelogram Examples

Solution:

Let ABCD be a parallelogram where DE⊥AB, AF⊥BC

Using the area of parallelogram formula,

Area of Parallelogram ABCD = (10) × (5) = 50 in²
Length of BC = 6 in
Length of Altitude AF = (50 ÷ 6) = 8.3 in

Answer: Area of the given parallelogram = 50 in²; Altitude = 8.3 in

How To Find Area of Parallelogram With Vectors?

Solved Examples

Q.1: Find the area of a parallelogram with a base is 10 cm and a height is 5 cm.

Solution: Given, Base = 10 cm, Height = 5 cm

Area of Parallelogram = B × H

Area = (10 cm) ͯ× (5 cm)

Area = 50 cm²

Therefore, Area of Parallelogram is 60 cm².

Q.2:Find the perimeter of a parallelogram whose base is 20 cm and height 12 cm?

Solution: Given,
Base = 20 cm
Height = 12 cm

Perimeter of a Parallelogram = 2(Base + Height)
= 2(20 + 12)
= 2 × 32 cm
= 64 cm

Therefore, Perimeter of a Parallelogram is 64 cm

Solved Examples For You

Question 1. The base and the corresponding altitude of a parallelogram are 10 cm and 3.5 cm respectively. The area of the parallelogram is:

1. 30  cm²
2. 35  cm²
3. 70  cm²
4. 17.5 cm²

Answer : B. The area of parallelogram is given by, base × height cm². Therefore, the area of the given parallelogram = 10 × 3.5 = 35 cm²

Question 2: The hypotenuse of a right-angled isosceles triangle is 5 cm The area of the triangle is:

1. 5 cm²
2. 6.25 cm²
3. 6.5 cm²
4. 12.5 cm²

Answer : B. Let the two sides be a² + a² = 25
2a² = 25
a² = 12.5

Area of a triangle = 1/2× a × a = a²/2
= 12.5/2
= 6.25 cm²

Question 3: What is the area and perimeter of a parallelogram?

Answer: The area A of a parallelogram is given by the formula. A=bh. Over here, b is the length of one base and h refers to the height. Further, the perimeter of a parallelogram is the sum of the lengths of its four sides.

Question 4: How do we find the perimeter of a triangle?

Answer: To calculate the perimeter of a triangle, add the length of its sides. For example, if a triangle has sides a, b, and c, then the perimeter of that triangle will be P = a + b + c.

Question 5: What are the diagonals of a parallelogram?

Answer: The diagonals bisect each other. One pair of opposite sides is parallel and equal in length. Adjacent angles are supplementary. Each diagonal divides the quadrilateral into two congruent triangles.

Question 6: What are the 5 properties of a parallelogram?

Answer: The five properties of a parallelogram are that both pairs of opposite sides are parallel and congruent. Moreover, both pairs of opposite angles are also congruent and the consecutive angles are supplementary. Finally, the diagonals of a parallelogram bisect each other.

Example 1

Calculate the area of a parallelogram whose base is 10 centimeters and height is 8 centimeters.

Solution

A = (b * h) Sq. units.

A = (10 * 8)

A = 80 cm²

Example 2

Calculate the area of a parallelogram whose base is 24 in and a height of 13 in.

Solution

A = (b * h) Sq. units.

= (24 * 13) square inch.

= 312 square inches.

Example 3

If the base of a parallelogram is 4 times the height and the area is 676 cm², find the parallelogram’s base and height.

Solution

Let the height of the parallelogram = x

and the base = 4x

But, the area of a parallelogram = b * h

676 cm² = (4x * x) Sq. units

676 = 4x²

Divide both sides by 4 to get,

169 = x²

By finding the square root of both sides, we get,

x = 13.

Substitute.

Base = 4 * 13 = 52 cm

Height = 13 cm.

Therefore, the base and height of the parallelogram are 52 cm and 13 cm, respectively.

Apart from the area of the parallelogram formula, there are other formulas for calculating the area of a parallelogram.

Let’s take a look.

How to find the area of a parallelogram without a height?

If the parallelogram’s height is unknown to us, we can use the trigonometry concept to find its area.

Area = ab sine (α) =ab sine (β)

Where a and b are the length of parallel sides, and either β or α is the angle between the sides of the parallelogram.

Example 4

Find the area of a parallelogram if its two parallel sides are 80 cm and 40 cm and the angle between them is 56 degrees.

Solution

Let a = 80 cm and b = 40 cm.

The angle between a and b = 56 degrees.

Area = ab sine (α)

Substitute.

A = 80 × 40 sine (56)

A = 3,200 sine 56

A = 2,652.9 sq.cm.

Example 5

Calculate the angles between the two sides of a parallelogram if its side lengths are 5 m and 9 m and the area of the parallelogram is 42.8 m².

Solution

Area of a parallelogram = ab sine (α)

42.8 m² = 9 * 5 sine (α)

42.8 = 45 sine (α)

Divide both sides by 45.

0.95111= sin (α)

α = sine^-1 0.95111

α = 72°

But β + α = 180°

β = 180° – 72°

= 108°

Therefore, the angles between the two parallel sides of the parallelogram are; 108° and 72°.

Example 6

Calculate the height of a parallelogram whose parallel sides are 30 cm and 40 cm, and the angle between these two sides is 36 degrees. Take the base of the parallelogram to be 40 cm.

Solution

Area = ab sine (α) = bh

30 * 40 sine (36) = 40 * h

1,200 sine (36) = 40 * h.

Divide both sides by 40.

h = (1200/40) sine 36

= 30 sine 36

h = 17.63 cm

So, the height of the parallelogram is 17.63 cm.

How to find the area of a parallelogram using diagonals?

Suppose d₁ and d₂ are the diagonals of the parallelogram ABCD, then the area of the parallelogram is given as,

A = ½ × d₁ × d₂ sine (β) = ½ × d₁ × d₂ sine (α)

Where β or α is the angle of intersection of the diagonals d₁ and d₂.

Example 7

Calculate the area of a parallelogram whose diagonals are 18 cm and 15 cm, and the angle of intersection between the diagonals is 43°.

Solution

Let d₁ = 18 cm and d₂ = 15 cm.

β = 43°.

A = ½ × d₁ × d₂ sine (β)

= ½ × 18 × 15 sine (43°)

= 135sine 43°

= 92.07 cm²

Therefore, the area of the parallelogram is 92.07 cm².

✅ Math Formulas ⭐️⭐️⭐️⭐️⭐

Gia sư toán bằng Tiếng Anh