**Revision Notes on Mensuration**

**Plane Figures**

The closed 2-D shapes are referred to as plane figures.

Here “C” is the boundary of the above figure and the area inside the boundary is the region of this figure. Point D comes in the area of the given figure.

**Perimeter**

If we go around the figure along its boundary to form a closed figure then the distance covered is the perimeter of that figure. Hence the Perimeter refers to the length of the boundary of a closed figure.

If a figure is made up of line segments only then we can find its perimeter by adding the length of all the sides of the given figure.

**Example**

Find the Perimeter of the given figure.

**Solution**

Perimeter = Sum of all the sides

= (12 + 3 + 7 + 6 + 10 + 3 + 15 + 12) m

= 68 m

**The Perimeter of a Rectangle**

A rectangle is a closed figure with two pairs of equal opposite sides.

Perimeter of a rectangle = Sum of all sides

= length + breadth + length + breadth

Thus, Perimeter of a rectangle = 2 × (length + breadth)

**Example: 1 **

The length and breadth of a rectangular swimming pool are 16 and 12 meters respectively .find the perimeter of the pool.

**Solution: **

Perimeter of a rectangle = 2 × (length + breadth)

Perimeter of the pool = 2 × (16 + 12)

= 2 × 28

= 56 meters

**Example: 2 **

Find the cost of fencing a rectangular farm of length 24 meters and breadth 18 meters at 8/- per meter.

**Solution:**

Perimeter of a rectangle = 2 × (length + breadth)

Perimeter of the farm = 2 × (24 + 18)

= 2 × 42

= 84 meter

Cost of fencing = 84 × 8

= Rs. 672

Thus the cost of fencing the farm is Rs. 672/-.

**Regular Closed Figure**

Figures with equal length of sides and an equal measure of angles are known as **Regular Closed Figures or Regular Polygon**.

Perimeter of Regular Polygon = Number of sides × Length of one side

**Perimeter of Square**

Square is a regular polygon with 4 equal sides.

Perimeter of square = side + side + side + side

Thus, Perimeter of a square = 4 × length of a side

**Example **

Find the perimeter of a square having side length 25 cm.

**Solution**

Perimeter of a square = 4 × length of a side

Perimeter of square = 4 × 25

= 100 cm

**Perimeter of an Equilateral Triangle**

An equilateral triangle is a regular polygon with three equal sides and angles.

Perimeter of an equilateral triangle = 3 × length of a side

**Example **

Find the perimeter of a triangle having each side length 13 cm.

**Solution**

Perimeter of an equilateral triangle = 3 × length of a side

Perimeter of triangle = 3 × 13

= 39 cm

**Perimeter of a Regular Pentagon **

A regular pentagon is a polygon with 5 equal sides and angles.

Perimeter of a regular pentagon = 5 × length of one side

**Example **

Find the perimeter of a pentagon having side length 9 cm.

**Solution**

Perimeter of a regular pentagon = 5 × length of one side

Perimeter of a regular pentagon = 5 × 9

= 45 cm

**Perimeter of a Regular Hexagon **

A regular hexagon is a polygon with 6 equal sides and angles.

Perimeter of a regular hexagon = 6 × Length of one side

**Example**

Find the perimeter of a hexagon having side length 15cm.

**Solution**

Perimeter of a regular hexagon = 6 × Length of one side

Perimeter of a regular hexagon = 6 × 15

= 90 cm

**Perimeter of a Regular Octagon **

A regular octagon is a polygon with 8 equal sides and angles.

Perimeter of a regular octagon = 8 × length of one side

**Example**

Find the perimeter of an octagon having side length 7cm.

**Solution**

Perimeter of a regular octagon = 8 × length of one side

Perimeter of a regular octagon = 8 × 7

= 56 cm

**Area**

Area refers to the surface enclosed by a closed figure.

To find the area of any irregular closed figure, we can put them on a graph paper with the square of 1 cm × 1 cm .then estimate the area of that figure by counting the area of the squares covered by the figure.

Here one square is taken as 1 sq.unit.

**Example**

Find the area of the given figure. (1 square = 1 m^{2})

**Solution**

The given figure is made up of line segments and is covered with some full squares and some half squares.

Full squares in figure = 32

Half squares in figure = 21

Area covered by full squares = 32 × 1 sq. unit = 32 sq. unit.

Area covered by half squares = 21 × (1/2) sq. unit. = 10.5 sq. unit.

Total area covered by figure = 32 + 10.5 = 42.5 sq. unit.

**Area of a Rectangle**

Area of a rectangle = (length × breadth)

**Example**

Find the area of a rectangle whose length and breadth are 20 cm and 12 cm respectively.

**Solution **

Length of the rectangle = 20 cm

Breadth of the rectangle = 12 cm

Area of the rectangle = length × breadth

= 20 cm × 12 cm

= 240 sq cm.

**To find the length of a rectangle if breadth and area are given:**

**Example**

What will be the length of the rectangle if its breadth is 6 m and the area is 48sq.m?

**Solution **

Length = 48/6

= 8 m

**To find the breadth of the rectangle if length and area are given:**

**Example **

What will be the breadth of the rectangle if its length is 8 m and the area is 81 sq.m?

**Solution**

Breadth = 81/8

= 9 m

**Area of a Square**

Area of a square is the region covered by the boundary of a square.

Area of a square = side × side

**Example**

Calculate the area of a square of side 13 cm.

**Solution**

Area of a square = side × side

= 13 × 13

= 169 cm^{2}.