Shapes with the Same Perimeter

In geometry, different shapes can sometimes have equal perimeters even if their sides or dimensions are completely different. This concept helps us understand how perimeter is not dependent on shape, but rather on the total length around the shape.

This topic is important in real-life scenarios like using the same piece of wire to create different shapes or comparing designs with the same boundary length (e.g., fencing).

The Fixed Boundary Principle states that if you use a fixed length of material (like a piece of string or wire) to form a boundary, the total length (the perimeter) remains constant regardless of the shape you create.

Problem: Each side of an equilateral triangle is 20 cm. Find the perimeter of the triangle.
A square has its perimeter same as that of the above triangle. Find the side of the square. 

Solution:

Each side of the given equilateral triangle = 20 cm

∴ Perimeter of the triangle = 3 × side

                                           = 3 × 20 cm = 60 cm

=> Perimeter of the square = 60 cm

=> 4 × the side of square = 60 cm

=> The side of the square = `"60 cm"/ "4"` = 15 cm

Problem: A wire is bent in the form of a square of side 25 cm.
Find the length of the wire.
If the same wire is bent in the form of a rectangle of length 30 cm, find the width of the rectangle.

Solution:

Length of the wire = 4 × side of the square

                                = 4 × 25 cm = 100 cm

Llength of the wire = perimeter of the rectangle

                                       = 2 × (length + breadth)

=> 100 cm = 2 × (30 cm + breadth)

=> 100 cm = 60 cm + 2 × breadth

=> 100 cm − 60 cm = 2 × breadth

=> 40 cm = 2 × breadth

=> breadth = `40/2` cm = 20 cm

Situation:

You are designing a border using beads or lace around a shape.

You use 36 cm of lace to border:

• A square photo frame (9 cm each side)
• A rectangular frame (13 cm × 5 cm)
• A triangular design (12 cm per side)

The lace length (perimeter) is the same, but the design layout varies.