Page 165 - Ganit Kaushal

Perimeter and Area of Closed Curves

The perimeter of any closed plane figure is the distance covered along its boundary and the amount of region
enclosed is called its area. Here we explain perimeter. Area will be explained in the next section.

For a polygon, i.e., a closed plane figure made up of line segments, the perimeter is simply the sum of the
lengths of all its sides, i.e., the total distance along its outer boundary. Thus, the following:

The perimeter of a polygon = Sum of lengths of all its sides

Perimeter of an Equilateral Triangle A

Let DABC be an equilateral triangle. Each side of DABC is 3 cm.
Now, Perimeter of DABC = Sum of all sides 3 cm 3 cm
= (3 + 3 + 3) cm = 9 cm = 3 × 3 cm = 3 × Length of a side
C 3 cm B
Hence, Perimeter of an equilateral triangle = 3 × (Length of a side)

For any general DABC, the perimeter is given as the sum of its sides.

Hence, Perimeter of DABC = AB + BC + CA

Perimeter of a Square

Let square ABCD have each side of length 2 m which is shown alongside. A 2 m B
Perimeter of square ABCD = Sum of all its sides = 2 m + 2 m + 2 m + 2 m = 8 m
2 m 2 m
= 4 × (2 m) = 4 × Length of a side
Hence, Perimeter of a square = 4 × (Length of a side) D 2 m C

Perimeter of a Regular Polygon

Analysing the formula of perimeter for an equilateral triangle and a square, which are regular polygons with
three sides and four sides respectively, we can generalise the formula for the perimeter of a regular polygon
with ‘N’ sides as follows:

Perimeter of ‘N’ sided regular polygon = N × (Length of a side)

Perimeter of a Rectangle

Consider a rectangle PQRS whose length (l) and breadth (b) are 12 cm and 8 cm, respectively.

Perimeter of the rectangle = Sum of the lengths of its four sides P 12 cm Q
= (12 + 8 + 12 + 8) cm = (24 + 16) cm 8 cm 8 cm

= 2(12 + 8) cm = 2 (l + b)
Hence, we conclude that: S 12 cm R

Perimeter of a rectangle = 2(Length + Breadth)

Perimeter and Area 163