Sachin

In Mensuration the circumference and area of a circle are defined as the length of the boundary of the circle and region occupied by the circle in 2-D Geometry. Let us discuss in detail the area and circumference of the circle using the formulas and solved example problems. We provide a detailed explanation of how to calculate the circumference and area of the circle.

What is Circumference and Area of Circle?

Circumference of Circle:

The circumference of the circle is the measure of the boundary of the circle. The circumference of the circle is also known as the perimeter of the circle. The perimeter or circumference of the circle is measured in units.
C = Πd or 2Πr

Area of Circle:

The area of the circle is the region covered by the circle or sphere in two-dimensional mensuration. The units to measure the area of the circle is square units.
A = Πr²
Where,
A is the area of the circle
r is the radius of the circle

What is the radius of the circle?

The radius of the circle is the distance from the center to the outline of the circle. Radius plays an important role in calculating the area and perimeter of the circle.

Properties of Circle

The properties of the circle are given below,

• The diameter of the circle is the longest chord of the circle.
• The circle is said to be congruent if it has the same radii.
• A circle can confine rectangle, square, trapezium, etc.
• If the tangents are drawn at the end of the diameter they are parallel to each other.

Area and Circumference of Circle Formula

Circumference:
The circumference of the circle is the measure of the boundary of the circle. The formula for the circumference of the circle is given below,
C = Πd
Where,
C is the circumference of the circle
Π is the mathematical constant
The approximate value of pi is 3.14 or 22/7
d is the diameter of the circle
C = 2Πr
Where,
C is the circumference of the circle
Π is the mathematical constant
The approximate value of pi is 3.14 or 22/7
r is the radius of the circle
Area: 
The formula for the area of the circle is as follows,
A = Πr²
A is an area of the circle
Π is the mathematical constant
The approximate value of pi is 3.14 or 22/7
r is the radius of the circle
Area of Semi-Circle:
The area of the semicircle is the region covered by the 2D figure. The formula for the area of the semi-circle is as follows,
A = Πr²/2
Perimeter of the Semi-circle:
The formula for the perimeter of the semi-circle is given below,
P = 2Πr/2 = Πr

Solved Examples on Circumference and Area of a Circle?

Get the step by step explanation on the formula of Area and Circumference of Circle here.

1. What is the circumference of the circle with a radius of 7cm?

Solution:

Given,
r = 7cm
We know that,
Circumference of the circle = 2Πr
C = 2 × 22/7 × 7 cm
C = 44 cm
Thus the circumference of the circle is 44 cm.

2. What is the circumference of the circle with a diameter of 21 cm?

Solution:

Given,
d = 21 cm
We know that,
Circumference of the circle = 2Πr
C = Πd
C = 22/7 × 21cm
C = 22 × 3cm
C = 66 cm
Therefore the circumference of the circle is 66 cm.

3. Find the area of the circle with a radius of 14m?

Solution:

Given,
r = 14m
We know that,
Area of Circle = Πr²
A = 22/7 × 14 × 14 sq.m
A = 22 × 2 × 14 m²
Thus the area of the circle is 616m²

4. Find the area of the circle if its circumference is 124m?

Solution:

Given,
Circumference of the circle = 124m
We know that,
Circumference of the circle = 2Πr
124m = 2Πr
Πr = 124/2
Πr = 62
r = 62 × 7/22
r = 19.72 m
Now find the area of the circle using the radius.
Area of Circle = Πr²
A = 3.14 × (19.72)²
A = 1221 sq. m
Therefore the area of the circle is 1221 sq. meters.

5. Find the area and circumference of the circle of radius 14m?

Solution:

Given,
radius = 14m
We know that,
Circumference of the circle = 2Πr
C = 2 × 22/7 × 14m
C = 2 × 22 × 2m
C = 88m
Now find the area of the circle using the radius.
Area of Circle = Πr²
A = Π(14)²
A = 22/7 × 14 × 14 sq.m
A = 22 × 2 × 14
A = 44 × 14 sq.m
A = 616 sq.m
Thus the area of the circle is 616 sq.m

FAQs on Circumference and Area of Circle

1. How to calculate the area of a circle?

The area of the circle can be calculated by the product of pi and radius squared.

2. What is the diameter of the circle?

The diameter of the circle is 2r.

3. How to calculate the circumference of the circle?

The circumference of the circle can be calculated by multiplying the diameter with pi.

The area of trapezium is the region covered by the trapezium in 2-D geometry. A trapezium is a type of quadrilateral that consists of four sides, four angles, and a set of parallel lines. The Formula Area of Trapezium is used in the concept of Mensuration in order to measure the two-dimensional figures. Let us learn the basic properties of trapezium from this article. Read this article thoroughly to know how to calculate the area of the trapezium with the help of the example problems. Also, learn the derivation of the area of the trapezium from here.

Area of Trapezium Definition

The area of trapezium is the region covered by the trapezium in two-dimensional geometry. The area of trapezium is equal to the product of half of the sum of parallel sides and distance between the parallel sides. The area of trapezium is measured in square units.

Properties of Trapezium

The properties of a trapezium are as follows,

• The sum of the four angles is equal to 360 degrees.
• The trapezium consists of a set of parallel sides and a set of non-parallel sides.
• The diagonals of the trapezium bisect each other.
• All the sides of the trapezium are not equal.

Area of Trapezium Formula

The formula for Area of Trapezium can be found by the product of half of the sum of parallel lines and distance between the parallel sides.

• Area of Trapezium = 1/2 (a + b)h

Derivation of Area of Trapezium Formula

Area of trapezium = sum of the area of rectangles and area of the triangle.
That means,
Area of trapezium = area of rectangle + area of triangle 1 + area of triangle 2
A = a1h + bh/2 + ch/2
A = (2a1h + bh + ch)/2
Take h as common
A = (a1 + a1 + b + c)h/2
A = (a1 + (a1 + b + c)h/2
Let a2 = a1 + b + c
A = (a1 + a2)h/2

Solved Examples on Area of the Trapezium

1. ABCD is a trapezium in which AB||CD, AD⊥DC, AB = 10 cm, DC = 15 cm and BC = 7 cm. Find the area of the trapezium?

Solution:

Given,
AB = 10 cm
DC = 15 cm
BC = 7 cm
We know that,
Area of Trapezium = 1/2 (a + b)h
PC = DC – DP
PC = 15 cm – 10 cm
PC = 5 cm
Now area of trapezium = Area of rectangle ABPD + Area of ΔBPC
Area of ΔBPC
BC² = BP² + PC²
7² = BP² + 5²
BP² = 49 – 25
BP² = 24
BP = 4.89 cm
Now area of trapezium ABCD = Area of rectangle ABPD + Area of ΔBPC
Area of rectangle ABPD = 10 × 4.89 = 48.9 sq. cm
Area of ΔBPC = 1/2 (5) (7) = 17.5 sq. cm
Area of trapezium ABCD = 48.9 + 17.5 = 66.4 sq. cm
Thus the area of trapezium ABCD is 66.4 sq. cm

2. Given height equal to 10cm, sides equal to 6cm and 5cm. Find the area of the trapezium?

Solution:

Given,
Height = 10cm
a = 6cm
b = 5cm
We know that,
Area of Trapezium = 1/2 (a + b)h
A = 1/2 (6cm + 5cm) 10cm
A = 11cm × 5cm
A = 55 sq. cm
Therefore the area of the trapezium is 55 sq. cm.

3. The area of the trapezium is 81 sq. m, sides are 9m and 6m. Find the height of the trapezium.

Solution:

Given,
The area of the trapezium is 81 sq. m
a = 9m
b = 6m
We know that,
Area of Trapezium = 1/2 (a + b)h
81 = 1/2 (9m + 6m)h
162 = 15m × h
h = 162/15 = 10.8m
Therefore the height of the trapezium is 10.8 meters.

FAQs on Area of Trapezium

1. What is the area of a trapezium?

The area of trapezium is half of the product of the height and the sum of the parallel lines.

2. What is trapezium?

A trapezium is a geometrical shape that has four sides and one set of parallel lines. It consists of four vertices and four angles.

3. What is the area of the trapezium formula?

Formula for Area of trapezium = 1/2 (a + b)h

The Area and Perimeter of the Rhombus are used in the basic Mensuration. In order to find the area and perimeter students must know what are the properties of the Rhombus. Here we explain the properties, formulas with examples. A rhombus is a quadrilateral which is similar to the parallelogram. The shape of the Rhombus looks like a diamond.

In this article, students can learn how to calculate the area and perimeter of the Rhombus. We have provided multiple examples to make the students understand the concept of Perimeter and Area of Rhombus. So, learn different methods to find the area and perimeter of the rhombus.

What is Area and Perimeter of a Rhombus?

Area of Rhombus – The area of the Rhombus is the space occupied by the two-dimensional figure. The Area of Rhombus Formula is equal to half of the product of two diagonals. The area of the rhombus is measured in square units.

• A = 1/2(d1 × d2) sq. units

where d1 and d2 are diagonals of the rhombus.

Perimeter of Rhombus – The perimeter of the rhombus is the sum of lengths of the boundaries. The Perimeter of the Rhombus Formula is equal to the sum of four sides. The perimeter of the Rhombus is measured in units.

• P = a + a + a + a = 4a units

Where a is the side of the rhombus.

Properties of Rhombus

The properties of the rhombus are given below

• All the sides of the rhombus are equal.
• It consists of 4 vertices and 4 edges.
• Opposite angles of the rhombus are the same.
• The sum of the adjacent angles is 180 degrees.
• In Rhombus the diagonals bisect the angles.
• Opposite sides of the rhombus are parallel.

Solved Examples on Area and Perimeter of a Rhombus

Go through the below section to know where and how to use the area and perimeter of the rhombus problems.

1. Find the area of the rhombus if the diagonals are 7m and 5m.

Solution:

Given,
d1 = 7m
d2 = 5m
We know that,
Area of the Rhombus = 1/2(d1 × d2)
A = 1/2(7m × 5m)
A = 35/2
A = 17.5 m²
Therefore the area of the rhombus is 17.5 sq. meters.

2. The area of the rhombus is 196 sq. cm. One of the diagonal is 14 cm find the other diagonal?

Solution:

Given,
The area of the rhombus is 169 sq. cm.
d1 = 14 cm
d2 = ?
We know that,
Area of the Rhombus = 1/2(d1 × d2)
196 sq. cm = 1/2 (14 cm × d2)
196 sq. cm = 7 cm × d2
d2 = 196/7 = 28 cm
Thus the length of the another diagonal is 28 cm.

3. What is the perimeter of the rhombus if the side is 6cm?

Solution:

Given,
a = 6 cm
We know that,
The perimeter of the Rhombus = 4a
P = 4(6 cm)
P = 24 cm
Thus the perimeter of the rhombus is 24 cm.

4. Find the area of the rhombus whose sides are 8 cm and diagonal is 6 cm?

Solution:

Given,

PQ = QR = RS = PS = 8 cm
QS = 6 cm
In ΔPOQ,
PQ² = OQ² + OP²
8² = 3² + OP²
64 = 9 + OP²
OP² = 64 – 9
OP² = 55
OP = 7.4
PR = 2OP
PR = 7.4 × 7.4
PR = 54.76
Area of the Rhombus = 1/2(d1 × d2)
A = 1/2 (6 × 54.76)
A = 164.28 sq. cm
Therefore the area of the rhombus is 164.28 sq. cm.

5. If the perimeter of the rhombus is 52 cm then find the side of the rhombus?

Solution:

Given,
the perimeter of the rhombus is 52 cm
We know that,
Perimeter of Rhombus = 4a
52 cm = 4a
a = 52/4
a = 13 cm
Thus the side of the rhombus is 13 cm.

6. Find the height of the rhombus whose area is 169 sq. m and perimeter is 140 m?

Solution:

Given that
P = 140 m
P = 4a
140 = 4a
a = 140/4
a = 35
Now use the area of the rhombus to find the height.
169 sq. m = 35 × h
h = 169/35
h = 4.82 m
Thus the height of the rhombus is 4.82 meters.

FAQs on Area and Perimeter of Rhombus

1. What are the basic properties of the Rhombus?

1. Opposite sides are equal
2. The sum of the adjacent angles is 180 degrees
3. All sides are equal in Rhombus.

2. How to find the perimeter of the rhombus?

The perimeter of the rhombus can be calculated by adding all four sides.

3. Is the area of the rhombus and square is the same?

The length of the rhombus is the same but the area of the square and rhombus are not equal.

A Parallelogram is a 2-D figure which is used in the Mensuration. The Area and Perimeter of the Parallelogram are similar to the area and perimeter of the rectangle. Learn the basic formulas of Parallelogram with the help of this article. Scroll down this page to know the formulas of Area and Perimeter of Parallelogram with solved problems.

Before learning the formulas we suggest the students to know what is a parallelogram. A parallelogram is a regular polygon with four sides and four angles. Get the properties, formulas from the below section.

Area and Perimeter of Parallelogram

Area of Parallelogram: The area of the parallelogram is the product of base and height. It is used to measure the region occupied by the parallelogram. The units to measure the area of the parallelogram is square units.

Perimeter of Parallelogram: The perimeter of the parallelogram is defined as the sum of all four sides. The units to measure the perimeter of the parallelogram is units.

Properties of Parallelogram

• The diagonals of the parallelogram bisect each other.
• Opposite sides of the parallelogram are congruent.
• If one of the angles is the right angle then all the angles are right.
• Opposite angles are congruent.

Formulas for Area and Perimeter of Parallelogram

The area of the parallelogram is the product of base and height.
Area = b × h
Where,
b = base
h = height
The perimeter of the parallelogram is the sum of the four sides.
P = a + a + b + b
P = 2a + 2b
P = 2(a + b)
Where,
a and b are the lengths of the parallelogram.

Solved Examples on Area of Parallelogram

1. Find the area of the parallelogram if the base and height are 12m and 10m?

Solution:

Given,
b = 12m
h = 10m
We know that,
Area of the Parallelogram = b×h
A = (12m)(10m)
A = 120 sq. meters
Therefore the area of the parallelogram is 120 sq.m

2. Find the perimeter of the parallelogram if the sides are 5 cm and 4 cm.

Solution:

Given,
a = 5 cm
b = 4 cm
We know that,
The perimeter of the parallelogram is 2(a + b)
P = 2(5 cm + 4 cm)
P = 2(9 cm)
P = 18 cm
Thus the perimeter of the parallelogram is 18 cm.

3. Find the height of the parallelogram if the base is 14m and Area is 112 sq. meters.

Solution: Given,
b = 14m
A = 112 sq. m
Area of the Parallelogram = b×h
112 sq. m = 14 × h
h = 112/14
h = 8 meters
Thus the height of the parallelogram is 8 meters.

4. Find the length of the parallelogram whose base length is 6m and the perimeter is 16m.

Solution:

Given,
a = 6m
b =?
P = 16m
Perimeter of the Parallelogram = 2(a + b)
16m = 2(6 + b)
16/2 = 6 + b
8m = 6m + b
b = 8m – 6m
b = 2m
Thus the length of the parallelogram is 2 meters.

FAQs on Area and Perimeter of the Parallelogram

1. What is the formula for area and perimeter of the parallelogram?

Area of the parallelogram = bh
Perimeter of the parallelogram = 2(a + b)

2. How to find the perimeter of the parallelogram?

The perimeter of the parallelogram can be found by adding the sides of the parallelogram i.e, a + a + b + b.

3. What is the height of the parallelogram?

The height of the parallelogram is the distance between the opposite sides of the parallelogram.

Area and Perimeter are the two important formulas in the basic theory of Mensuration. This article helps to learn the properties and types of triangles here. The two main concepts to measure the sides of the triangle are area and perimeter. There are different formulas to find the perimeter and area of the triangle.

A two-dimensional closed figure with three sides is called a triangle. Our main aim is to make the students understand the concept of Area and Perimeter of the Triangle. Learn how to calculate the area and perimeter of the triangle by using the below mentioned solved examples.

What is the Area and Perimeter of the Triangle?

Area: Area is defined as the measure of the region occupied by the triangle. The units to measure the area of the triangle is square meters or square centimeters.

Perimeter: The perimeter is the measure of lengths of the triangle. The perimeter of the triangle is the sum of all the sides of the triangle. The units to measure the perimeter of the triangle is meters or centimeters.

Area and Perimeter of the Triangle Formula

The area of the triangle is half of the base and height.
A = 1/2 × base × height
The perimeter of the triangle is the sum of three sides of the triangle.
P = a + b + c

Area of Triangle with three sides (Heron’s Formula)

The Area of the triangle with three sides can be found using Heron’s Formula. There are two steps to find the area of the triangle using Heron’s Formula. The first step is to find the value of s (semiperimeter of the triangle) by adding all the three sides i.e, a, b, c, and dividing by 2. The next steps is to apply the semi-perimeter of the triangle with the main formula

s = (a+b+c)/2
A = √s(s-a)(s-b)(s-c)

Types of Triangles

There are 4 types of triangles. They are:

• Equilateral triangle
• Isosceles triangle
• Scalene triangle
• Right triangle

Properties of triangle

• The sum of three angles of the triangle is 180º
• A triangle has three sides, three vertices, and three angles.
• The sum of two lengths of the triangle is greater than the third side.
• The area of the triangle is half of the base and height.

Worked Out Example Problems on Area of the Triangle

1. Find the area of the triangle whose base is 10 cm and height is 11 cm?

Solution:

Given,
b = 10 cm
h = 11 cm
Area of triangle = 1/2 × b × h
A = 1/2 × 10 cm × 11 cm
A = 5 cm × 11 cm
A = 55 sq. cm
Therefore the area of the triangle is 55 sq. cm

2. Find the missing length whose perimeter is 36cm and two sides of the triangles are 14 cm?

Solution:

Given,
a = 14 cm
b = 14 cm
c = x
P = 36 cm
Perimeter of the triangle = a + b + c
36 cm = 14 cm + 14 cm + x
36 cm = 28 cm + x
x = 36 cm – 28 cm
x = 8 cm
Thus the length of third side of the triangle is 8 cm.

3. Given a = 10m, b = 12m and c = 13m. Find the perimeter of the scalene triangle?

Solution:

Given,
a = 10m
b = 12m
c = 13m
Perimeter of the triangle = a + b + c
P = 10m + 12m + 13m
P = 35m
Thus the perimeter of the scalene triangle is 35 meters.

4. The area of the triangle is 144 cm² and the base is 12 cm. Find the height of the triangle?

Solution:

Given,
The area of the triangle is 144 cm²
base = 12 cm.
Area of triangle = 1/2 × b × h
144cm² = 12 cm × h
h = 144cm²/12cm
h = 12 cm
Thus the height of the triangle is 12 cm.

5. Find the area of the isosceles triangle whose base is 7m and height is 10m?

Solution:

Given,
b = 7m
h = 10m
Area of the isosceles triangle = 1/2 × b × h
A = 1/2 × 7m × 10m
A = 7m × 5m
A = 35 sq. m
Thus the area of the isosceles triangle is 35 sq. m.

6. Find the semi perimeter of the triangle whose sides of the triangle a = 5cm, b = 6cm, and c = 7cm using Heron’s formula?

Solution:

Given,
a = 5cm
b = 6cm
c = 7 cm
s = (a+b+c)/2
s = (5cm + 6cm + 7 cm)/2
s = 18cm/2
s = 9 cm
Therefore the semi-perimeter of the triangle is 9cm.

7. Find the area of the triangle with three given sides a = 1m, b = 3m, c = 4m?

Solution:

Given,
a = 2m
b = 3m
c = 4m
First, we need to find the semiperimeter of the triangle.
s = (a+b+c)/2
s = (1 + 3 + 4)/2
s = 8/2
s = 4m
Now find the area of the triangle with three sides
A = √s(s-a)(s-b)(s-c)
A = √4(4-1)(4-3)(4-4)
A = √4(3)(1)(0)
A = 0

FAQs on Area and Perimeter of the Triangle

1. What is the formula for area and perimeter of the triangle?

The area of the triangle is 1/2 × b × h
The perimeter of the triangle is a + b + c

2. How to calculate the perimeter of the triangle?

We can calculate the perimeter of the triangle by adding all three sides.
P = a + b + c

3. What is the relationship between the area and perimeter of the triangle?

The area is half of the base and height whereas the perimeter is the sum of all the three sides of the triangle.

Path is nothing but the measure of boundaries of the shapes. The area of the path is defined as the region occupied by the shape inside or outside. Here we can apply the concept of Area of rectangle or area square to find the area of paths. This is the major concept in the basic theory of Mensuration. The students can learn the different techniques to find the area of the path of different shapes like rectangle, square, circle, etc.

What is the Area of Path?

The area of the path is the subtraction of the area of the outer rectangle/square and the area of the inner rectangle/square. It is defined as the space left between the area of two 2-d figures or paths. This article helps to learn the theory of the area of the path in depth.

Area of the Path Formula

Area of the path = area of outer rectangle – area of inner rectangle
Area of the path = area of outer square – area of inner square

Solved Problems on Area of the Path

Here we use the concept of the area of rectangle and area of the square. Know how to calculate the area of the path with the help of the solved example problems.

1. A rectangular plot is 20 cm long and 15 cm wide. The inner rectangular plot is 12 cm long and 10 cm wide. Draw the diagram of the rectangular plot and find the area of the path?

Solution:

Given the length of the outer rectangle is 20 cm
Width of the outer rectangle is 15 cm
We know that,
Area of the rectangle = l × w
A = 20 cm × 15 cm
A = 300 sq. cm
Also given the length of the inner rectangle = 12 cm
Width of the inner rectangular plot = 10 cm
We know that,
Area of the rectangle = l × w
A = 12 cm × 10 cm
A = 120 sq. cm
Total Area of the path = 300 sq. cm – 120 sq. cm = 180 sq. cm

2. A park is of length 30 m and 20 m wide. A path 1.5 m wide is constructed outside the rectangular garden. Find the area of the path?

Solution:

Given the length of the inner rectangle = 30 meter
The wodth of the inner rectangle = 20 meter
We know that,
Area of the rectangle = l × w
A = 30 m × 20 m
A = 600 sq. meter
AB = 20 + 1.5 + 1.5 = 23 m
BC = 30 + 1.5 + 1.5 = 33 m
Area of the outer rectangle = l × w
A = 33 m × 23 m
A = 759 sq. meter
Area of the path = Area of outer rectangle – Area of inner rectangle
Therefore Area of the path = 759 sq. m – 600 sq. m
A = 159 square meters
Thus the area of the path of rectangular garden is 159 sq. meters.

3. A path 10 m wide runs along inside the square park of side 1000 m. Find the area of the path? 

Solution:

Given that,
A path 10 m wide runs along inside the square park of side 1000 m.
PS = PQ = QR = RS = 1000m
Find the sides of the inner square.
Width = 10m
AB = 1000 – 10 – 10 = 980m
AB = BC = CD = AD = 980 m
We know that
Area of the square = side × side
A = 1000 × 1000 = 1,000,000 m²
Area of the inner square = s × s
A = 980 × 980 = 960,400 m²
Area of the path = 1,000,000 m² – 960,400 m²
= 39,600m²
Thus the area of the path is 39,600m²

4. A rectangular box is 10 cm long and 8 cm wide. A path of the uniform width is 5 cm. Find the area of path?

Solution:
Given,
A rectangular box is 10 cm long and 8 cm wide. A path of the uniform width is 5 cm
Find the length and width of the outer rectangle
AD = 10cm + 5cm + 5cm = 20cm
AB = 8cm + 5cm + 5cm = 18cm
Area of the outer rectangle = l × w
A = 20cm × 18cm
A = 360 sq. cm
Area of the inner rectangle = l × w
A = 10cm × 8cm
A = 80 sq. cm
Area of the path = Area of outer rectangle – Area of inner rectangle
A = 360 sq. cm – 80 sq. cm
A = 280 sq. cm
Thus the area of path is 280 sq. cm

5. A 1m wide path runs outside and around a rectangular park of length 6m and width 4m. Find the area of the path?

Solution:
Given,
A 1m wide path runs outside and around a rectangular park of length 6m and width 4m.
Area of inner rectangle = l × w
A = 6m × 4m
A = 24 sq. m
Area of outer rectangle = (6m + 1m + 1m) × (4m + 1m + 1m)
A = 8m × 6m
A = 48 sq. m
Area of the path = Area of the outer rectangle – Area of inner rectangle
A = 48 sq. m – 24 sq. m
A = 24 sq. meters
Thus the area of the path is 24 sq. meters.

FAQs on Area of the Path

1. How to find the area of the path?

We can find the area of a path by subtracting the area of outer shapes and the area of inner shapes.

2. What is the formula for Area of Path?

Area of the path = Area of outer rectangle/square – Area of inner rectangle/square

3. How to calculate the area of the path for square?

You can calculate the Area of the path using the formula Area of Path = Area of outer square – Area of inner square

The Perimeter and Area of the Square are used to measure the length of the boundary and space occupied by the square. These are two important formulas used in Mensuration. Perimeter and Area of the Square formulas are used in the 2-D geometry.

Square is a regular quadrilateral where are the sides and angles are equal. The concepts of the Perimeter and Area Square formula, Derivation, Properties, are explained here. The solved examples with clear cut explanations are provided in this article. Students can understand how and where to use the formulas of Area and Perimeter of Square.

What is the Area and Perimeter of the Square?

Area of a square: The area of the square is defined as the region covered by the two-dimensional shape. The units of the area of the square are measured in square units i.e., sq. cm or sq. m.

Perimeter of a square: The perimeter of the square is a measure of the length of the boundaries of the square. The units of the perimeter are measured in cm or m.

Area of Square Formula

The area of the square is equal to the product of the side and side.
Area = Side ×  Side sq. units
A = s² sq. units

Perimeter of Square Formula

The perimeter of the square is the sum of the lengths.
P = s + s + s +s
P = 4s units
Where s is the side of the square.

Diagonal of Square Formula

The square has two diagonals with equal lengths. The diagonal of the square is greater than the sides of the square.

• The relationship between d and s is d = a√2
• The relationship between d and Area is d = √2A

What is Square?

A square is a regular polygon in which all four sides are equal. The measurement of the angles of the square is also equal.

Properties of Squares

The properties of the square are similar to the properties of the rectangle. Go through the properties of squares from the below section.

• All sides of the squares are equal.
• It has 4 sides and 4 vertices.
• The interior angles of the square are equal to 90º
• The diagonlas of square bisect at 90º
• The diagonals of the square are divided into two isosceles triangles.
• The opposite sides of the squares are parallel to each other.
• Each half of the square is equal to two rectangles.

Solved Problems on Perimeter and Area of Square

Below we have provided the solved examples of perimeter and area of a square with a brief explanation. Scroll down this page to check out the formulas of Area and Perimeter of Square.

1. What is the Area and Perimeter of the square if one of its sides is 4 meters?

Solution:

Given the side of the square is 4 meters.
Area of the square = s × s
A = 4 m × 4 m
A = 16 sq. meters
The perimeter of the square = 4s
P = 4 × 4 m
P = 16 meters.
Therefore the area and perimeter of the square are 16 sq. meters and 16 meters.

2. Find the area of the square if the side is 10 cm?

Solution:

Given,
s = 10 cm
Area of the square = s × s
A = 10 cm × 10 cm
A = 100 sq. cm
Therefore the area of the square is 100 sq. cm

3. The perimeter of the square is 64 cm. Find the area of the square?

Solution:

Given,
The perimeter of the square is 64 cm
P = 4s
64 cm = 4s
s = 64/4 = 16 cm
Thus the side of the square is 16 cm.
Now the find the area of the square.
Area of the square = s × s
A = 16 cm × 16 cm
A = 256 sq. cm
Therefore the area of the square is 256 sq. cm.

4. If the area of the square is 81 cm², then what is the length of the square?

Solution:

Given,
A = 81 cm²
Area of the square = s × s
81 sq. cm = s²
s² = 81 sq. cm
s = √81 sq. cm
s = 9 cm
Thus the length of the square is 9 cm.

5. The length of the square is 25 cm. What is the area of the square?

Solution:

Given,
The length of the square is 25 cm
Area of the square = s × s
A = 25 × 25
A = 625 sq. cm
Therefore the area of the square is 625 sq. cm.

FAQs on Perimeter and Area of Square

1. How to find the perimeter of the square?

Add all the sides of the square to find the perimeter of the square.

2. What is the formula for the perimeter of a square?

The Perimeter of Square formula is sum of the lengths i.e, side + side + side + side = 4s

3. What is the formula for the area of the square?

The area of the square formula is the product of side and side. A = s × s.

The Perimeter and Area of Rectangle are two important formulas in Mensuration. It calculates the space occupied by the rectangle and the length of boundaries of the rectangle. In this article, students can come across the concept of area and perimeter of rectangle deeply.

A rectangle is a quadrilateral with two equal sides and two parallel lines and four right angles. The concept of area and perimeter of rectangle formulas are explained with examples. Some of the examples of different shapes are given below.

What is Perimeter and Area of a Rectangle?

Perimeter: Perimeter of the rectangle is the sum of all the sides of the rectangle. The rectangle has two lengths and two breadths. To find the perimeter of the rectangle we have to add the length and breadth. It is measured in units. It is denoted by P.

Area: The area of the rectangle formula helps to calculate the length and breadth of the two-dimensional closed figure. To find the area of the rectangle we have to multiply the length and breadth of the rectangle. It is measured in square units. It is denoted by A.

Properties of Rectangle

• A Rectangle has two equal sides
• The rectangle is a quadrilateral
• The diagonals of the rectangle have the same length.
• The diagonal of the rectangle bisect each other.
• Sum of all the four angles is 360º
• Each angle of the rectangle is 90º
• If the sides of the rectangle are l and b then the diagonal of the rectangle d = √l² + b²
• The opposite sides of the rectangle are parallel.

Derivation of Perimeter of a Rectangle

The perimeter of the rectangle is the sum of all four sides.
P = l + l + b + b
P = 2l + 2b
P = 2(l + b)
Thus the perimeter of the rectangle is = 2(l + b)

Area of the Rectangle

The area of the rectangle is the product of length and breadth.
A = l × b

Perimeter and Area of Rectangle Formula

• Area of Rectangle = l × b
• Perimeter of Rectangle = 2(l + b)
• Length of the rectangle = A/b
• The breadth of the Rectangle = A/l
• Diagonal of the Rectangle = √l² + b²

Solved Problems on Area and Perimeter of a Rectangle

The formula of Perimeter and Area of Rectangle is explained step by step here with examples. Go through the below questions and solve the problems using the Area and Perimeter of the Rectangle formula.

1. Find the length of the rectangular plot whose breadth is 11 cm and the area is 165 cm². Also, find the perimeter of the rectangle?

Solution:

Given,
Breadth = 11 cm
Area = 165 cm²
Area of the rectangle = l × b
165 sq. cm = l × 11 cm
l = 165/11
l = 15 cm
Thus the length of the rectangle is 15 cm.
We know that,
Perimeter of the rectangle = 2(l + b)
P = 2(15 + 11)
P = 2(26)
P = 52 cm
Thus the perimeter of the rectangle is 52 cm.

2. Find the Perimeter of the Rectangle whose length is 10 cm and breadth is 8 cm?

Solution:

Given, Length = 10 cm
Breadth = 8 cm
We know that,
Perimeter of the rectangle = 2(l + b)
P = 2(10 cm + 8 cm)
P = 2(18 cm)
P = 36 cm
Therefore the perimeter of the rectangle is 36 cm.

3. Find the area of the rectangle whose length is 14 meter and width is 10 meters?

Solution:

Given,
Length = 14 meter
Width = 10 meter
We know that,
Area of the rectangle = l × w
A = 14 m × 10 m
A = 140 sq. meters
Therefore the area of the rectangle is 140 square meters.

4. A rectangular plot has its length of 16 cm and a perimeter of 60 cm. Find the width of the rectangular plot?

Solution:

Given,
Length = 16 cm
Perimeter = 60 cm
Width = ?
We know that
The perimeter of the rectangle = 2(l + w)
60 cm = 2(16 cm + w)
16 + w = 60/2
16 + w = 30
w = 30 – 16
w = 14 cm
Thus the width of the rectangular plot is 14 cm.

5. Find the area and perimeter of the rectangle whose length and breadth are 12 m and 6 m?

Solution:
Given,
length = 12 m
breadth = 6 m
We know that,
Area of the rectangle = l × b
A = 12 m × 6 m
A = 72 sq. meters
Now find the perimeter of the rectangle
P = l + l + b + b
P = 12 m + 12 m + 6 m + 6 m
P = 24 m + 12 m
P = 36 m
Therefore the area and perimeter of the rectangle is 72 square meters and 36 meters.

FAQs on Area and Perimeter of Rectangle

1. How to find the perimeter of a rectangle?

The perimeter of the rectangle can be calculated by adding all the sides of the rectangle.

2. What is the area of the rectangle?

The area of the rectangle is defined as the space occupied by the rectangle or closed figure. It is the product of length and breadth.

3. What is the unit for the perimeter of a rectangle?

The unit for the perimeter of the rectangle is cm or meters.

Area and Perimeter is an important and basic topic in the Mensuration of 2-D or Planar Figures. The area is used to measure the space occupied by the planar figures. The perimeter is used to measure the boundaries of the closed figures. In Mathematics, these are two major formulas to solve the problems in the 2-dimensional shapes.

Each and every shape has two properties that are Area and Perimeter. Students can find the area and perimeter of different shapes like Circle, Rectangle, Square, Parallelogram, Rhombus, Trapezium, Quadrilateral, Pentagon, Hexagon, and Octagon. The properties of the figures will vary based on their structures, angles, and size. Scroll down this page to learn deeply about the area and perimeter of all the two-dimensional shapes.

Area and Perimeter Definition

Area: Area is defined as the measure of the space enclosed by the planar figure or shape. The Units to measure the area of the closed figure is square centimeters or meters.

Perimeter: Perimeter is defined as the measure of the length of the boundary of the two-dimensional planar figure. The units to measure the perimeter of the closed figures is centimeters or meters.

Formulas for Area and Perimeter of 2-D Shapes

1. Area and Perimeter of Rectangle:

• Area = l × b
• Perimeter = 2 (l + b)
• Diagnol = √l² + b²

Where, l = length
b = breadth

2. Area and Perimeter of Square:

• Area = s × s
• Perimeter = 4s

Where s = side of the square

3. Area and Perimeter of Parallelogram:

• Area = bh
• Perimeter = 2( b + h)

Where, b = base
h = height

4. Area and Perimeter of Trapezoid:

• Area = 1/2 × h (a + b)
• Perimeter = a + b + c + d

Where, a, b, c, d are the sides of the trapezoid
h is the height of the trapezoid

5. Area and Perimeter of Triangle:

• Area = 1/2 × b × h
• Perimeter = a + b + c

Where, b = base
h = height
a, b, c are the sides of the triangle

6. Area and Perimeter of Pentagon:

• Area = (5/2) s × a
• Perimeter = 5s

Where s is the side of the pentagon
a is the length

7. Area and Perimeter of Hexagon:

• Area = 1/2 × P × a
• Perimeter = s + s + s + s + s + s = 6s

Where s is the side of the hexagon.

8. Area and Perimeter of Rhombus:

• Area = 1/2 (d1 + d2)
• Perimeter = 4a

Where d1 and d2 are the diagonals of the rhombus
a is the side of the rhombus

9. Area and Perimeter of Circle:

• Area = Πr²
• Circumference of the circle = 2Πr

Where r is the radius of the circle
Π = 3.14 or 22/7

10. Area and Perimeter of Octagon:

• Area = 2(1 + √2) s²
• Perimeter = 8s

Where s is the side of the octagon.

Solved Examples on Area and Perimeter

Here are some of the examples of the area and perimeter of the geometric figures. Students can easily understand the concept of the area and perimeter with the help of these problems.

1. Find the area and perimeter of the rectangle whose length is 8m and breadth is 4m?

Solution:

Given,
l = 8m
b = 4m
Area of the rectangle = l × b
A = 8m × 4m
A = 32 sq. meters
The perimeter of the rectangle = 2(l + b)
P = 2(8m + 4m)
P = 2(12m)
P = 24 meters
Therefore the area and perimeter of the rectangle is 32 sq. m and 24 meters.

2. Calculate the area of the rhombus whose diagonals are 6 cm and 5 cm?

Solution:

Given,
d1 = 6cm
d2 = 5 cm
Area = 1/2 (d1 + d2)
A = 1/2 (6 cm + 5cm)
A = 1/2 × 11 cm
A = 5.5 sq. cm
Thus the area of the rhombus is 5.5 sq. cm

3. Find the area of the triangle whose base and height are 11 cm and 7 cm?

Solution:

Given,
Base = 11 cm
Height = 7 cm
We know that
Area of the triangle = 1/2 × b × h
A = 1/2 × 11 cm × 7 cm
A = 1/2 × 77 sq. cm
A = 38.5 sq. cm
Thus the area of the triangle is 38.5 sq. cm.

4. Find the area of the circle whose radius is 7 cm?

Solution:

Given,
Radius = 7 cm
We know that,
Area of the circle = Πr²
Π = 3.14
A = 3.14 × 7 cm × 7 cm
A = 3.14 × 49 sq. cm
A = 153.86 sq. cm
Therefore the area of the circle is 153.86 sq. cm.

5. Find the area of the trapezoid if the length, breadth, and height is 8 cm, 4 cm, and 5 cm?

Solution:

Given,
a = 8 cm
b = 4 cm
h = 5 cm
We know that,
Area of the trapezoid = 1/2 × h(a + b)
A = 1/2 × (8 + 4)5
A = 1/2 × 12 × 5
A = 6 cm× 5 cm
A = 30 sq. cm
Therefore the area of the trapezoid is 30 sq. cm.

6. Find the perimeter of the pentagon whose side is 5 meters?

Solution:

Given that,
Side = 5 m
The perimeter of the pentagon = 5s
P = 5 × 5 m
P = 25 meters
Therefore the perimeter of the pentagon is 25 meters.

FAQs on Area and Perimeter

1. How does Perimeter relate to Area?

The perimeter is the boundary of the closed figure whereas the area is the space occupied by the planar.

2. How to calculate the perimeter?

The perimeter can be calculated by adding the lengths of all the sides of the figure.

3. What is the formula for perimeter?

The formula for perimeter is the sum of all the sides.

In Maths Mensuration is nothing but a measurement of 2-D and 3-D Geometrical Figures. Mensuration is the study of the measurement of shapes and figures. We can measure the area, perimeter, and volume of geometrical shapes such as Cube, Cylinder, Cone, Cuboid, Sphere, and so on.

Keep reading this page to learn deeply about the mensuration. We can solve the problems easily, if and only we know the formulas of the particular shape or figure. This article helps to learn the mensuration formulae with examples. Learn the difference between the 2-D and 3-D shapes from here. Understand the concept of Mensuration by using various formulas.

• Area and Perimeter
• Perimeter and Area of Rectangle
• Perimeter and Area of Square
• Area of the Path
• Area and Perimeter of the Triangle
• Area of Parallelogram
• Area of Rhombus
• Area of Trapezium
• Circumference and Area of Circle
• Area Conversion
• Practice Test on Area and Perimeter of Rectangle
• Practice Test on Area and Perimeter of Square

Definition of Mensuration

Mensuration is the theory of measurement. It is the branch of mathematics that is used for the measurement of various figures like the cube, cuboid, square, rectangle, cylinder, etc. We can measure the 2 Dimensional and 3 Dimensional figures in the form of Area, Perimeter, Surface Area, Volume, etc.

What is a 2-D Shape?

The shape or figure with two dimensions like length and width is known as the 2-D shape. An example of a 2-D figure is a Square, Rectangle, Triangle, Parallelogram, Trapezium, Rhombus, etc. We can measure the 2-D shapes in the form of Area (A) and Perimeter (P).

What is 3-D Shape?

The shape with more or than two dimensions such as length, width, and height then it is known as 3-D figures. Examples of 3-Dimensional figures are Cube, Cuboid, Sphere, Cylinder, Cone, etc. The 3D figure is determined in the form of Total Surface Area (TSA), Lateral Surface Area (LSA), Curved Surface Area (CSA), and Volume (V).

Introduction to Mensuration

The important terminologies that are used in mensuration are Area, Perimeter, Volume, TSA, CSA, LSA.

• Area: The Area is an extent of two-dimensional figures that measure the space occupied by the closed figure. The units for Area is square units. The abbreviation for Area is A.
• Perimeter: The perimeter is used to measure the boundary of the closed planar figure. The units for Perimeter is cm or m. The abbreviation for Perimeter is P.
• Total Surface Area: The total surface area is the combination or sum of both lateral surface area and curved surface area. The units for the total surface area is square cm or m. The abbreviation for the total surface area is TSA.
• Lateral Surface Area: It is the measure of all sides of the object excluding top and base. The units for the lateral surface area is square cm or m. The abbreviation for the lateral surface area is LSA.
• Curved Surface Area: The area of a curved surface is called a Curved Surface Area. The units of the curved surface area are square cm or m. The abbreviation for the curved surface area is CSA.
• Volume: Volume is the measure of the three dimensional closed surfaces. The units for volume is cubic cm or m. The abbreviation for Volume is V.

Mensuration Formulas for 2-D Figures

Check out the formulas of 2-dimensional figures from here. By using these mensuration formulae students can easily solve the problems of 2D figures.

1. Rectangle:

• Area = length × width
• Perimeter = 2(l + w)

2. Square:

• Area = side × side
• Perimeter = 4 × side

3. Circle:

• Area = Πr²
• Circumference = 2Πr
• Diameter = 2r

4. Triangle:

• Area = 1/2 × base × height
• Perimeter = a + b + c

5. Isosceles Triangle:

• Area = 1/2 × base × height
• Perimeter = 2 × (a + b)

6. Scalene Triangle:

• Area = 1/2 × base × height
• Perimeter = a + b + c

7. Right Angled Triangle:

• Area = 1/2 × base × height
• Perimeter = b + h + hypotenuse
• Hypotenuse c = a²+b²

8. Parallelogram:

• Area = a × b
• Perimeter = 2(l + b)

9. Rhombus:

• Area = 1/2 × d1 × d2
• Perimeter = 4 × side

10. Trapezium:

• Area = 1/2 × h(a + b)
• Perimeter = a + b + c + d

11. Equilateral Triangle:

• Area = √3/4 × a²
• Perimeter = 3a

Mensuration Formulas of 3D Figures

The list of the mensuration formulae for 3-dimensional shapes is given below. Learn the relationship between the various parameters from here.

1. Cube:

• Lateral Surface Area = 4a²
• Total Surface Area = 6a²
• Volume = a³

2. Cuboid:

• Lateral Surface Area = 2h(l + b)
• Total Surface Area = 2(lb + bh + lh)
• Volume = length × breadth × height

3. Cylinder:

• Lateral Surface Area = 2Πrh
• Total Surface Area = 2Πrh + 2Πr²
• Volume = Πr²h

4. Cone:

• Lateral Surface Area = Πrl
• Total Surface Area = Πr(r + l)
• Volume = 1/3 Πr²h

5. Sphere:

• Lateral Surface Area = 4Πr²
• Total Surface Area = 4Πr²
• Volume = (4/3)Πr³

6. Hemisphere:

• Lateral Surface Area = 2Πr²
• Total Surface Area = 3Πr²
• Volume = (2/3)Πr³

Solved Problems on Mensuration

Here are some questions that help you to understand the concept of Mensuration. Use the Mensuration formulas to solve the problems.

1. Find the Length of the Rectangle whose Perimeter is 24 cm and Width is 3 cm?

Solution:

Given that,
Perimeter = 24 cm
Width = 3 cm
Perimeter of the rectangle = 2(l + w)
24 cm = 2(l + 3 cm)
2l + 6 = 24
2l + 6 = 24
2l = 24 – 6 = 18
2l = 18
l = 9 cm
Thus length of the rectangle = 9 cm

2. Calculate the volume of the Cuboid whole base area is 60 cm² and height is 5 cm.

Solution:

Given,
Base area = 60 cm²
Height = 5 cm
Volume of the Cuboid = base area × height
V = 60 cm² × 5 cm
V = 300 cm³
Thus the volume of the cuboid is 300 cm³.

3. Find the area of the Cube whose side is 10 centimeters.

Solution:

Given, side = 10 cm
Lateral Surface Area = 4a²
LSA = 4 × 10 × 10 = 400 cm²
Total Surface Area = 6a²
= 6 × 10 × 10 = 600 cm²
Volume of the cube = a³
V = 10 × 10 × 10 = 1000 cm³
Therefore the volume of the cube is 1000 cubic centimeters.

4. What is the lateral surface area of the sphere if the radius is 5 cm.

Solution:

Given,
The radius of the sphere = 5 cm
The formula for LSA of sphere = 4Πr²
Π = 3.14 or 22/7
LSA = 4 × 3.14 × 5 cm × 5 cm
LSA = 314 sq. cm
Thus the lateral surface area of the sphere is 314 sq. cm

5. What is the area of the parallelogram if the base is 15 cm and height is 10 cm.

Solution:

Given, Base = 15 cm
Height = 10 cm
We know that,
Area of parallelogram = bh
A = 15 cm × 10 cm
A = 150 sq. cm
Therefore the area of the parallelogram is 150 sq. cm.

FAQs on Mensuration

1. What is the use of Mensuration?

Mensuration is used to find the length, area, perimeter, and volume of the geometric figures.

2. What is the difference between 2D and 3D figures?

In 2D we can measure the area and perimeter. In 3D we can measure LSA, TSA, and Volume.

3. What is Mensuration in Math?

Mensuration is the branch of mathematics that studies the theory of measurement of 2D and 3D geometric figures or shapes.