Perimeter of Closed Figures: Definitions, Explanation, Examples

The word perimeter comes from two Greek words: peri, which means “around,” and metron, which means “measure.” The perimeter of a closed figure is defined as the path or boundary that encircles the shape in geometry. Each shape’s perimeter varies depending on its measurements. The perimeter of a circle is only stated as the circle’s diameter. However, calculating the perimeter of all polygons is the same: we must sum all of their sides.

Closed figures are those geometrical figures bounded by lines straight or curved like polygons and circles. In this article, we will learn the various ways to find the perimeter of the polygons like rectangle, square, triangles, quadrilateral, etc., and the perimeter of irregular closed figures.

Concept of Perimeter

We know that the concept of perimeter applies to rectilinear figures. A figure formed by joining the straight lines or segments is called a rectilinear figure. These lines are called the sides of the figure. Examples of rectilinear figures are triangles, squares, rectangles, quadrilaterals, and polygons.

Consider each of the figures given above. In each case, start moving from \(O\) along the line segments to reach again at the same point \(O\) by making a complete round, then the distance covered in making one full round is the perimeter of the figure.

Thus, the perimeter of a rectilinear figure is the distance around its boundary. Or we can say that the total length of the boundary of a closed shape or rectilinear figure is called its perimeter. So, the sum of the lengths of all sides of a rectilinear figure is its perimeter. The perimeter measurement unit is the unit of length, like centimeter \((\mathrm{cm})\) meter \(\left( {\rm{m}} \right),\) etc. 

Perimeter of Closed Figures

As of now, we know the basic concept of finding the perimeter. Let us look out the perimeter of various closed figures in detail. Let us start with polygons.

Perimeter of Triangles

\(\Delta ABC\) is any triangle with sides \(a,b,\) and \(c.\)

So, the perimeter of \(\Delta ABC=\) Sum of the length of the sides \(=A B+B C+C A=c+a+b=a+b+c .\)

Perimeter of Some Special Types of Triangles

Equilateral Triangle

If \(\Delta ABC\) is an equilateral triangle with each side a, then

The perimeter of \(\Delta ABC=\) the length of the sides \(=AB+BC+CA=a+a+a=3a\)

\( = 3 \times {\rm{side}}\)

Isosceles Triangle

If \(\Delta ABC\) is any triangle with \(AB=AC=a\) and \(BC=b\)

So, the perimeter of \(\Delta ABC=\) the length of the sides \(=AB+BC+CA=a+b+a=2a+b\)

Perimeter of a Quadrilateral

A quadrilateral is a \( 4-\)sides closed figure.

If \(a, b, c,\) and \(d\) are the lengths of the four sides of a quadrilateral, then its perimeter \(=AB+BC+CD+DA=a+b+c+d.\)

Perimeter of Some Special Types of Quadrilaterals

Perimeter of a Rectangle: A rectangle is a quadrilateral whose opposite sides are equal and parallel to each other. The sum of all the sides of a rectangle is called its perimeter. The figure below shows a rectangle \(ABCD\) in which,

Side \(AB =\)side \(DC = l\) (length of the rectangle)

And, Side \(BC=\) side \(AD=b\) (breadth of the rectangle)

Therefore, the perimeter of the rectangle \(=\) sum of the length of its sides

\(=AB+BC+CD+DA\)

\(=l+b+l+b\)

\(=2l+2b=2(l+b).\)

Perimeter of a Square

A square is a rectangle whose length is equal to breadth, i.e., all the sides of a square are of equal length.

Therefore, the perimeter of the square \(ABCD\)

\(=AB+BC+CD+DA\)

\(l+l+l+l=4l\)

\( = 4 \times {\rm{side}}\)

Perimeter of a Circle

All the points of a closed two-dimensional figure in which the plane are equidistant from a centre are called a circle. The distance around a circular region is known as the circumference of the circle. Thus, the perimeter of a circle is nothing but its circumference.

So, the perimeter of a circle with radius \(r\) is given by \(2 \pi r\) units.

Perimeter of a Semi-Circle

The perimeter of a semi-circle is \(=\frac{1}{2} \times 2 \pi r \times 2 r\)

\(=(\pi+r) r\)

Perimeter of a Quadrant of a Circle

The perimeter of a quadrant is \(=\frac{1}{4} \times 2 \pi r \times 2 r\)

\(=\left(\frac{\pi}{2}+r\right) r\)

Perimeter of Some Closed Figures

In our day-to-day life, we encounter shapes that are not in the form of quadrilaterals, like, rectangle, square, rhombus, parallelogram, kite, etc., or circular.  We have learnt to find the perimeter of these. But some shapes are closed shapes but not in the form of the shapes mentioned above; those shapes are known as irregular closed shapes. Then how to find the perimeter of those figures???

We know that sum of all the sides gives us the perimeter of the figure. Here we will apply the same formula. Let us solve a couple of examples to understand the concept better.

Example 1: Find the perimeter of the below-given shape.

Solution: The required perimeter

\(=3.4 \mathrm{~cm}+5.5 \mathrm{~cm}+1.4 \mathrm{~cm}+3 \mathrm{~cm}+1 \mathrm{~cm}+6 \mathrm{~cm}+2 \mathrm{~cm}+2.5 \mathrm{~cm}\)

\(=24.8 \mathrm{~cm}\)

Hence, the required perimeter is \(24.8 \mathrm{~cm}\)

Example 2: Find the perimeter of the given shape.

Solution: The sum of all the sides will give us the perimeter.

Thus, Perimeter \(=10 \mathrm{~cm}+4 \mathrm{~cm}+2 \mathrm{~cm}+3 \mathrm{~cm}+4 \mathrm{~cm}+3 \mathrm{~cm}+2 \mathrm{~cm}+4 \mathrm{~cm}\)

\(=32 \mathrm{~cm}\)

Hence, the perimeter is equal to \(32 \mathrm{~cm}\)

Example 3: Find the perimeter of the following given figure.

Answer: The perimeter \(=11 \mathrm{~cm}+2 \mathrm{~cm}+5 \mathrm{~cm}+4 \mathrm{~cm}+6 \mathrm{~cm}+6 \mathrm{~cm}\)

\(=34 \mathrm{~cm}\)

Hence, the required perimeter is \(34 \mathrm{~cm}\)

Shapes of Different Kinds With the Same Perimeter

Consider an equilateral triangle with each side \(=12 \mathrm{~cm}\)

The perimeter of the triangle \( = 3 \times {\rm{side}}\)

\(=3 \times 12 \mathrm{~cm}\)

\(=36 \mathrm{~cm}\)

Now, consider a rectangle with length \(=13 \mathrm{~cm}\) and breadth \(=5 \mathrm{~cm}\)

Its perimeter \( = 2{\rm{(length + breadth)}}\)

\(=2(13 \mathrm{~cm}+5 \mathrm{~cm})\)

\(=2 \times 18 \mathrm{~cm}=36 \mathrm{~cm}\)

Further, consider a square of side \(9 \mathrm{~cm}\)

Its perimeter \( = 4 \times {\rm{side}}\)

\(=4 \times 9 \mathrm{~cm}\)

\(=36 \mathrm{~cm}\)

In the example above, we find that even if the dimensions (length, breadth, etc.) of different figures are not equal, their perimeters are the same.

Consider the following shapes, which have sides of different lengths but the same perimeter.

Solved Examples

Q.1. Find the perimeter of each of the following closed figures;

a)

b)

Ans: a) The required perimeter \(=AB+BC+CD+DE+EF+FG+GA\)
\(=6.2 \mathrm{~cm}+4 \mathrm{~cm}+8 \mathrm{~cm}+4 \mathrm{~cm}+8 \mathrm{~cm}+4 \mathrm{~cm}+6.2 \mathrm{~cm}\)
\(=40.4 \mathrm{~cm}\)
Hence, the perimeter of the given figure is \(40.4 \mathrm{~cm}\)
b) The required perimeter \(=AB+BC+CD+DE+EF+FG+GH+HI+IJ+JK+KL+LA\)
\(=6 \mathrm{~cm}+6 \mathrm{~cm}+3 \mathrm{~cm}+6 \mathrm{~cm}+6 \mathrm{~cm}+3 \mathrm{~cm}+6 \mathrm{~cm}+6 \mathrm{~cm}+3 \mathrm{~cm}+6 \mathrm{~cm}\)
\(+6 \mathrm{~cm}+3 \mathrm{~cm}\)
\(=60 \mathrm{~cm}\)
Hence, the perimeter of the given figure is \(60 \mathrm{~cm}\)

Q.2. A wire is bent in the form of a square of side \(25\,{\rm{cm}}{\rm{.}}\) Find the length of the wire. If the same wire is bent in the form of a rectangle of length \(30\,{\rm{cm}}{\rm{,}}\) find the width of the rectangle.
Ans: Length of the wire \(=4×\) side of the square
\(=4 \times 25=100 \mathrm{~cm}\)
Now, the length of the wire=the perimeter of the rectangle
\( = 2{\rm{(length + breadth)}}\)
\(100\;{\rm{cm}} = 2 \times (30\;{\rm{cm}} + {\rm{breadth)}}\)
\( \Rightarrow 50\;{\rm{cm}} = 30\;{\rm{cm}} + {\rm{breadth}}\)
\(\Rightarrow {\rm{breadth = }}50 – 30\;{\rm{cm}}\)
\({\rm{Breadth = }}20\;{\rm{cm}}\)
Hence, the width of the rectangle is \(20\;{\rm{cm}}.\)

Q.3. A rectangle \(10\,{\rm{cm}}\) long has the same perimeter as a square of side \(8\,{\rm{cm}}.\) Find the breadth of the rectangle.
Ans: Side of a square \(=8 \mathrm{~cm}\)

The perimeter of the square \( = 4 \times {\rm{side}}\)
\(=4 \times 8 \mathrm{~cm}=32 \mathrm{~cm}\) 
We have the perimeter of rectangle \(=\) perimeter of square

So, \({\rm{2(length + breadth) = 32}}\)
\(\Rightarrow 2(10+b)=32\)
\(\Rightarrow 10+b=16\)
\(\Rightarrow b=16-10=6 \mathrm{~cm}\)
Thus the breadth of the rectangle is \(6 \mathrm{~cm}.\)

Q.4. Preethu makes \(4\) full rounds of a square field of side \(100\,{\rm{m}}.\) Priya also makes full \(4\) rounds of a rectangular field \(150\,{\rm{m}}\) long and \(55\,{\rm{m}}\) wide. Find who covers larger distance and by how much?
Ans: Distance covered by Preethu in one full round=perimeter of the square field
\( = 4 \times {\rm{side}} = 4 \times 100\,{\rm{cm}}\)
\(=400 \mathrm{~cm}\)
Distance covered by Preethu in \(4\) full rounds \( = 4 \times 400\,{\rm{m}}\)
\( = 1600\,{\rm{m}}\)
Distance covered by Priya in one full round=perimeter of the rectangular field
\({\rm{ = 2(length + breadth)}}\)
\(=2 \times(150+55) \mathrm{m}\)
\(=2 \times 205 \mathrm{~m}=410 \mathrm{~m}\)
Distance covered by Priya in \(4\) full rounds \( = 4 \times 410\,{\rm{m}}\)
\( = 1640\,{\rm{m}}\)
Priya covers a larger distance and by \(1640\,{\rm{m}} – 1600\,{\rm{m}} = 40\,{\rm{m}}\)

Q.5. The radius of a circular plate is \(7\,{\rm{m}}.\) Find the circumference of the plate. (Take \(\pi=\frac{22}{7}\))
Ans: 

Circumference of a circle \(=2 \pi r\)
\(=2 \times \frac{22}{7} \times 7\)
\(=2 \times 22 \mathrm{~m}\)
\(=44 \mathrm{~m}\)
Hence, the circumference of the circle is \(44\,{\rm{m}}.\)

Summary

In this article, first, we learnt about rectilinear figures or closed figures. Then we learnt the concept of perimeter, and followed by this, we learnt the formulas of finding the perimeter of various closed rectilinear figures. In addition to this, we also learnt that the figures with different dimensions could have the same perimeter. Lastly, we solved some examples to memorize the concept of perimeter better.

Learn About Area of Closed Figures

FAQs

Q.1. How do you find the perimeter of a closed figure?
Ans: The perimeter of a closed figure \(=\) the sum of the length of its sides.

Q.2. How do you find the perimeter of a curved shape?
Ans: The perimeter of a curved shape is the length of a rope held all around it.

Q.3. What is the perimeter formula?
Ans: Perimeter is the distance covered along the boundary forming a closed figure when you go round the figure once. The perimeter of a shape is defined as the overall distance around the shape. Thus, it is the length of any shape that can be expanded in a linear form.
So, the perimeter formula depends on the shape and measurement of the given shape or object.
Example: Perimeter of a square with the length of each side \(a\) units is given by \(4a\) units.
The perimeter of a rectangle with length \(l\) units  and breadth \(b\) units is given by \(2\left( {l + b} \right)\) units.

Q.4. What are the area and perimeter?
Ans: The area of a closed plane figure is the measurement of the region or surface enclosed by its boundary or sides.
The perimeter of a closed plane figure is the distance around its boundary. Or we can say that the total length of the boundary of a closed shape or rectilinear figure is called its perimeter.

Q.5. How do you find the perimeter of different figures?
Ans: We can find out the perimeter of different figures by finding the sum of the lengths of all sides.

We hope this detailed article on the perimeter of closed figures helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!