Perimeter and Area

 Mind Maps

Class V - maths: Perimeter, Area and Volume
Q) What is the amount of space taken by a solid object?

Q) What is the volume of a cuboid?

Q) What is the area of triangle?

Q) What is the perimeter of a square?

Q) What is the perimeter of a rectangle?

Q) Area of square = ?

Q) Find the side of the square whose perimeter is 40 cm?

Q) Area of rectangle = ?

Q) The perimeter of a figure is the total distance around the?

Q) The sum of all the sides of a square is its?

Q) Volume is the amount of?

Q) A cube is a cuboid having?

Q) Find the perimeter of a rectangle whose length is 7 cm and breadth is 2.5 cm?

Q) Find the side of a square whose perimeter is 32cm?

Q) Find the area of a lawn of length 25 m and breadth 15 m?

Q) Find the volume of a cuboid of length 8 cm, breadth 4 cm, and height 6 cm?

Q) Find the volume of a cube of side 4 cm?

Q) What is Perimeter?

Q) What is volume?

Q) What is Whole Squares?

Q) Mrs Nehra bought a carpet measuring 8.5 m by 7 m. What is the area of the carpet?

Q) The length and breadth of a cuboid are 6 cm and 5 cm, respectively. If the volume of the cuboid is 108 cubic cm, find the height of the cuboid?

Q) How many ice cubes of edge 2 cm can fit into an ice tray of measure 2 cmr4 cmr9cm?

Q) Find the area of rectangle with length 3cm and breadth 8 cm?

Q) Find the area of square with side 12 m?

Q) What is Perimeter and give an example?

Q) What is volume and give an example?

Q) What is Area Of Figures that do not Cover Whole Squares? Give an example?

Q) Find the length of the wire required to fence 3 rounds of a rectangular ground of length 20 m and breadth 15 m. Also find the cost of fencing the ground at the rate of ? 2 permetre?

### PERIMETER

Definition

"The perimeter of a figure is the total distance around the edge of the figure".
Example:

#### Perimeter By Using A Formula

The perimeter of some polygons can be calculated directly by using a formula.
Let us calculate the perimeter of a rectangle and square by using a formula.

### PERIMETER OF A RECTANGLE

In a rectangle, the perimeter is the sum of all its sides.

Therefore, perimeter        = l + b + l + b
= l + l + b + b
= 2× l + 2 × b
= 2 (l + b)
Perimeter of a rectangle = 2 ( length + breadth )

Examples:

1. Find the perimeter of a rectangle whose length is 5 cm and breadth is 4 cm.
Solution:

Here, length of the rectangle = 5 cm
Breadth of the rectangle = 4 cm
Perimeter of the rectangle = 2 (length + breadth)
= 2 (5 cm + 4 cm)
= 2 (9 cm)
= 18 cm.
Hence, the perimeter of the rectangle is 18 cm.

2. Find the length of the wire required to fence 3 rounds of a rectangular ground of length 20 m and breadth 15 m. Also find the cost of fencing the ground at the rate of ₨ 2 per metre.
Solution: Length of the ground = 20 m
Breadth of the ground = 15 m
Perimeter = 2 (length + Breadth)
= 2 (20 + 15)
= 2(35)
= 70m
Total wire required to fence 3 rounds of ground
= 3 × 70 m = 210 m
Cost of fencing 1m = ₨ 2
Cost of fencing 210 m = 210 × 2
Cost of fencing = ₨ 420

### PERIMETER OF A SQUARE

"The sum of all the sides of a square is its perimeter".

Therefore, Perimeter of a square = 4s = 4 × side
Examples:
1. Find the perimeter of a square whose side is 13 cm.
Solution:

Side of a square = 13 cm
Perimeter of a square = 4 × side
= 4 × 13 cm
= 52 cm.

2. Find the side of a square whose perimeter is 36 cm.
Solution: Perimeter of the square = 36 cm
Perimeter = 4 × side
36 = 4 × side
Side = 36 ÷ 4
Side = 9 cm.

### PERIMETER OF A TRIANGLE

"The sum of measure of three sides of a triangle is its perimeter".

So, Perimeter of a triangle, P = a + b + c
Example: Find the perimeter of a triangle whose sides are 11 cm, 13 cm, and 8 cm
Solution:

Perimeter of a triangle = sum of its sides
= 11 + 13 + 8
= 32 cm
So, the perimeter of the triangle = 32 cm.

### AREA

Definition:

"The amount of surface covered by a figure is known as the area of the figure".
It is always measure in 'square units'.
Example:

The figure above shows a rectangle of length 10 cm and breadth 6 cm.
The rectangle is divided into a number of squares of side 1 cm
The rectangle is covered with 60 such squares.
Therefore, the area of rectangle is 60 sq cm.
"The number of units needed to cover a surface is called its area".

#### Units Used For The Measurement Of Area

• Small surfaces are measured in square centimetres or cm2.
Example: Surface area of a book.
• Large surfaces are measured in square metres or m2.
Example: Surface area of a school.
• Still larger surfaces are measured in square kilometres or km2.
Example: Surface area of a city.

Finding area by counting squares is not always applicable; in such cases formulae are used.

### AREA OF RECTANGLE

Formula For Measuring Area Of A Rectangle

In the above figure, the rectangle is divided into squares of side 1 cm.
The area of each square is 1 sq cm.
Area of rectangle = number of unit squares in the rectangle
The area of rectangle = 18 sq cm
We can see from the above figure:
Number of square along the length = 6
Number of square along the breadth = 3.
On multiplying the two, we get 6 × 3 = 18 sq cm.
Area of Rectangle = length × breadth

Examples:
1. Find the area of a park of length 35 m and breadth 10 m.
Solution: Length of the park = 35 m
Breadth of the park = 10 m
Area of the park = (35 × 10) sq cm = 350 sq cm.

2. Find the breadth of a rectangle whose area is 160 sq cm and length is 20 cm.
Solution: Area of rectangle = 160 sq cm
Length = 20 cm
So, Area = length × breadth
= 8 cm.
So, the breadth of the rectangle = 8 cm.

### AREA OF A SQUARE

#### Formula For The Area Of A Square

In a square, all the sides are equal. This means the length and breadth are equal.

Area of a square = side × side
Area of a square = (side)2
Examples:

1. Find the area of a square whose side is 15 cm.
Solution:

Side of a square = 15 cm.
Area of square = side × side
= (15 × 15) sq cm
= 225 sq cm.

1. Find the area of a square field whose perimeter is 84 m.
Solution: Perimeter of the square field = 84 m
Perimeter = 4 × side
Side of the square field = perimeter ÷ 4
Side of the square field = 84/4 = 21 m
Area of the field = side × side
= 21 × 21 = 441 sq cm.

### AREA OF A TRIANGLE

Area of a triangle = 1/2 of the area of rectangle
(Or)
Area of a triangle = 1/2 × base × height.

Example:

1. Find the area of a triangle, with height 5 cm and base 10 cm.
Solution:

Area of the triangle = 1/2 × base × height
= 1/2 × 10 × 5
= 1/2 × 50
=25 sq cm
So, area of the triangle = 25 sq cm.

### Area Of Figures That Do Not Cover Whole Squares

There are some shapes which do not cover the whole squares or complete squares.
To find the area of such figures, we have to consider half squares besides complete squares.
Example:

The above figure covers 4 complete squares and 4 half squares.
We can treat 2 half squares as 1 whole square.
So, area of the figure = 4 + 2 = 6 sq units.

#### Estimation Of Area Of Irregular Shapes

There are many figures around us which do not have a perfect shape.
In order to calculate the area of such irregular shapes, we first calculate the estimate area of that particular figure.
For estimation of area, the following steps are followed:

• Count the complete whole squares
• Count the half squares
• Estimate the partial squares to calculate the number of complete squares that can be made.
• Add all the three calculations. The sum is the required approximate area.

Example:

In the above irregular figure,
Number of complete squares = 48
Number of half squares = 12 = 6 whole squares
Total area = 48 + 6 = 54 squares
As the area of each square is 1 cm2, the area of the irregular shape is 54 cm2.

### VOLUME

Definition

"Volume is the amount of space taken by a solid object".
The standard unit of volume is 1 unit, like 1mm, 1 cm, or 1m.
It is always measured in cubic units.
Example:

We can find the volume of solid figures by fitting in unit cubes in the solid and counting them.

#### Units Used For The Measurement Of Volume

• For very small containers, cubes of side 1 mm are used. The measurement of volume is written as 1 cubic mm or 1 mm3.
• For small containers, cubes of side 1 cm are used. The measurement of volume is written as 1 cubic cm or 1 cm3.
• For large containers, cubes of side 1 m are used. The measurement of volume is written as 1 cubic m or 1 m3.
VOLUME OF A CUBOID

To find the volume of a cuboid, we count the unit cubes.

If length, breadth and height of a cuboid are known, the volume can be calculated as:
Volume of a cuboid = length × breadth × height
Examples:
1. Find the volume of a cuboid whose length is 3 m, breadth is 2 m and height is 5 m.
Solution:

Volume of the cuboid = l × b × h
= 30 cubic metres.
1. The length and breadth of a cuboid are 8 cm and 4 cm respectively, if the volume of the cuboid is 192 cubic cm, find the height of the cuboid.
Solution: Volume of the cuboid = l × b × h
192 = 8 × 4 × h
192 = 32 × h
Height = 192 ÷ 32
Height = 6 cm.

VOLUME OF A CUBE

"A cube is a cuboid having equal length, breadth and height".

So, The volume of a cube = side × side × side
Examples:
1. Find the volume of a cube of side 5 cm
Solution:

Volume = side × side × side
= 5 × 5 × 5
= 125 cubic cm.

1. How many bricks of edge 3 cm can fit into a wall of measure 12 cm × 6 cm × 9 cm?
Solution: Volume of wall = 12 × 6 × 9 = 648 cm3
Side of a brick = 3 cm
Volume of the brick = 3 × 3 × 3
= 27 cm3
Number of bricks = Volume of the wall ÷ Volume of the brick
= 648/27
= 24.
So, 24 bricks can fit into the wall.

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