# Sides of polygons having the same perimeter

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In this lesson, we solve certain type of problems where we find side length of a polygon which has same perimeter as the given polygon.

Consider for an example: A wire is first bent in the shape of a rectangle of length 13 cm and 5 cm. Then this wire is unbent and reshaped into a square. We are now required to find the side length of this square.

It is clear that the length of the wire is fixed. The perimeter of the rectangle is the perimeter of the square. So we first find the perimeter of given rectangle using the formula 2(l + w). Since the rectangle is reshaped into a square, the perimeter of square is same as the perimeter of the rectangle.

Since all sides of a square are of equal length,

Side length of the square = $\frac{Square\:perimeter}{4}$ = $\frac{2(l + w)}{4}$

If the rectangle were reshaped into an equilateral triangle then the perimeter of the triangle would be same as the perimeter of the rectangle.

Since all sides of an equilateral triangle are of same length,

the side length of the equilateral triangle = $\frac{2(l + w)}{3}$

A wire is first bent into the shape of a rectangle with width 7 cm and 13 cm length. Then the wire is unbent and reshaped into a square. What is the length of a side of the square?

### Solution

Step 1:

Perimeter of rectangle = 2(7 + 13) = 40 cm

Step 2:

Perimeter of square = 40 cm

Side length of the square = $\frac{40}{4}$ = 10 cm

A wire is first bent into the shape of a rectangle with width 12 cm and 18 cm length. Then the wire is unbent and reshaped into a triangle. What is the length of a side of the triangle, if all its sides are equal?

### Solution

Step 1:

Perimeter of rectangle = 2(12 + 18) = 60 cm

Step 2:

Perimeter of equilateral triangle = 60 cm

Side length of the equilateral triangle = $\frac{60}{3}$ = 20 cm