Perimeter

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Perimeter

What is perimeter?

The total length of the boundary of a closed shape is called its perimeter. Let’s try and understand this using an example,

Perimeter of the shape \(= 5\,m + 8\,m + 4\,m + 6\,m = 15\,m\)

Sub Topics

Introduction to Perimeter of a Circle

Imagine a circular slice of pizza. If you know the radius (or diameter) of the pizza, you can easily find its perimeter and area. This was just one example, but there are several situations that need you to know the area and perimeter of a given circle.

What is pi ?

Before jumping into the technicalities, let’s familiarize ourselves with the diameter and the perimeter or circumference of a circle. To visualise the circumference, think of the crust of a Pizza, well, that’s essentially what it is! The length of the outer edge of any circle.

Here’s where things get really interesting. What if you were told that if you divide the circumference of ANY circle with its diameter, you will ALWAYS end up with the SAME number, 3.1415…

The use of ellipses is deliberate! This particular ratio is always the same regardless of the circle in question! Yes, it is mind boggling, which is why this number occupies such a crucial role in all of the mathematics related to the circle. This ratio is called Pi.

Regarding the number, pi is a non-repeating, non-terminating entity which is best approximated to the fraction \({22 \over 7}\). The decimal representation up to 10 places is 3.1428571429.

How is it important?

The perimeter of a circle

Like we understood for closed polygons bounded by straight lines, the perimeter of a circle is the length of the boundary enclosing that circular space. If you imagine a perfectly circular field with a white fence all around it, then the white fence can be said to represent the perimeter of the circular field. If you think of a pizza, then the outer crust of the pizza can be said to be its perimeter. For a circular space of radius \({r}\), the perimeter is calculated as

\({\text{Perimeter}} = 2 \times \pi \times r = 2\pi r\)

We do know that the radius of a circle is half its diameter, and if \({\text{D}}\) represents the diameter of the circle, then \({\text{D} = 2r}\). If we replace \({2r}\) of the perimeter formula by \({\text{D}}\), then,

\(\begin{align}{\text{Perimeter}} = \pi D\end{align}\)

If you have purchased a 14-inch pizza, you now know how to calculate the length of the outer boundary of the pizza.

\(\begin{align}{\text{Perimeter}} = \pi D = \frac{{22}}{7} \times 14 = 44{\text{ inches}}\end{align}\)

Let’s have a look at the circumference of the circle visually, so that the concept can be more easily internalized.

Circumference of a circle

Formula for calculating the perimeter of different shapes

Shape	Formula for perimeter	Examples
	\(2\,(\text{length} + \text{breadth})\)		\(\begin{align}P &= 2\,(4 \,cm + 2\, cm) \\ &= 2\, (6\, cm)\\ &= 12 \,cm\end{align}\)
	\(4 \times \text{side}\)		\(\begin{align}P &= 4 \times 7 \,cm \\ & = 28 \,cm \end{align}\)
	\(\begin{align} 2 \pi r \end{align}\)		\(\begin{align}P & = 2 \pi r \\ & = 2 \times { 22 \over 7} \times 14\\ & = 88 \,cm \end{align}\)
	\(a+b+c\)		\(\begin{align}P &= 5\,cm + 5 \,cm + 6 \,cm \\ &= 10 \,cm + 6 \,cm \\ &= 16 \,cm \end{align}\)

Tips and tricks

Tip: Children often get confused between area and perimeter as concepts. A thorough understanding can be built by asking the child to trace the surface of any 3D shape and measure its outside boundary. And then finding out the perimeter.
While calculating the perimeter of composite shapes children often miss summing up one of the sides to avoid this ask the children to calculate the horizontal shapes first followed by the vertical shapes. In the given figure calculating the perimeter of shape 1, followed by shape 2 and then shape 3 would ensure that all the sides are accounted for.

For rectilinear shapes doubling the sides doubles the perimeter. For example, the perimeter of the given shape is 2 (l + b) = 2 (40 cm + 20cm) = 120 m. However, if the sides double up (length 80m and width 40m) the perimeter will also double up, it will be 140m.