Answer sheets of meritorious students of class 12th’ 2012 M.P Board – All Subjects

February 14, 201339 Insightful Publications

**Perimeter and Area of Circle: **Circle is a very important geometric figure. In our daily life, we see many circular objects around us like a ring, bottom of a bottle, a steering wheel, etc. The length of the boundary of the circle is called the perimeter or circumference of a circle. The space or region enclosed by a circle is called the area of the circle.

Archimedes, a Greek mathematician, is credited with devising the method for calculating the value of pi. To approximate, he drew a regular polygon within the circle, and the more sides he drew on the polygon, the better the approximation he would achieve. The ratio of a circle’s circumference (perimeter) to its diameter had been known for a long time. A mathematical constant is a number. The ratio of the circumference to the diameter of any circle is. 3.14159 is a close approximation. The interactive below shows how to calculate pi’s value.

In this article, we will discuss in detail about the perimeter and area of a circle and more.

The locus of the set of all points in a plane, which is at equal distances from a fixed point, is called a circle. The circle is a two-dimensional figure.

The fixed point is called the centre of the circle, and the fixed distance is named the radius of the circle. The different terms associated with the circle are discussed below:

**Center**

The fixed point of the circle, from where all the points are drawn with equal distance, is known as the centre of a circle.**Radius**

The fixed distance from the point on the circle to the centre of a circle is called the radius of the circle. We can say that the line segment joining the centre to the point on the circle is called the radius of the circle.**Diameter**

The line segment joining two points on the circle, which passes through the centre of a circle, is called the diameter of the circle.**Chord**

The chord of a circle is the line segment joining any two points on the circumference of a circle.**Tangent**

A line that touches the circle at only one point is called the tangent of a circle.**Secant**

A line that intersects a circle at two points is called the secant of a circle.**Sector:**

The region enclosed by two radii and an arc of a circle is called the sector of a circle.**Segment**

The region formed by a chord and an arc of a circle is called a segment of a circle.**Arc**

Arc is a continuous part of the circumference of the circle.

**Circumference (Perimeter)**

Circumference of a circle or perimeter of a circle is the measure of the length of the boundary of the circle.

If the circle is cut at one end and opened to form a line segment, then its length is the circumference. It is generally measured in units like \({\rm{cm}}\) or \({\rm{m}}\).

**Area**

It is the space or region occupied by a circle in a two-dimensional plane.

When an arc is rotated around \({\rm{36}}{{\rm{0}}^{\rm{o}}}\) then the circumference of a circle is formed. If the arc is more than half of the circumference, then it is a major arc. If it is less than half of the circumference, it is a minor arc.

The diameter divides the circle exactly into two equal parts, and each part is known as a semi-circle.

Minor arc, Semi-circle and Major arcs are shown in the above figures in red colour.

Let us consider a circle with radius \(‘r’\). There are some formulas associated with the area and perimeter of a circle, which explained below:

In case the radius of a circle is given, then the circumference of the circle can be calculated by using the formula,

Where\(r\) represents the radius of the circle and \(\pi \) is the mathematical constant whose value is approximately equal to \(\frac{{22}}{7}\) or \(3.14\).

We know that diameter of a circle is equal to twice the radius of the circle, numerically. \((d = 2r)\)

**The circumference of a circle **\( = \pi \times {\rm{diameter}} = \pi \times d\)**, **where \(d\) represents the diameter of the circle.

The area of a circle with radius \(\left( r \right)\) is given by \({\rm{Area}} = \pi \times {({\rm{radius}})^2} = \pi {r^2}\)**.**

Here\(r\) represents the radius of the circle and \(\pi \) is the mathematical constant whose value is approximately equal to \(\frac{{22}}{7}\) or \(3.14\). (it is an irrational number with non-terminating decimal places).

We know that the diameter of a circle equals twice the radius of the circle, numerically. \((d = 2r)\)

**Area** \(\pi \times {\left( {\frac{d}{2}} \right)^2} = \frac{{\pi {d^2}}}{4}\)**, **where \(d\) represents the diameter of the circle.

The methods used to derive the area of a circle are:

- Using the area of a rectangle
- Using the area of a triangle.

In this method, the circle is divided into many small sectors, and the arrangement of the sectors has been made in the form of a parallelogram, as shown in the figure below.

Clearly, if the circle is divided into more sectors, the parallelogram will turn into a rectangle.

As observed in the figure, the circle is divided into \(16\) sectors; \(8\) of them are coloured blue, and the rest \(8\) are coloured green.

The blue and the green sectors are arranged alternately, as shown in the figure above.

Considering that the circumference of a circle is \(2\pi r\), the total length of the base of the parallelogram (rectangle as the number of sectors becomes more) will be \(\pi r\) (the total length of the bases of \(8\) blue sectors) and height of the parallelogram (or rectangle) will be \(r\).

So, the area of the rectangle \({\rm{ = length \times base}} = \pi r \times r = \pi {r^2}\)

The rectangle consists of the area of the sectors of the circle.

Hence, the area of the circle is \(\pi {r^2}\).

In this method, the entire area of the circle is considered as consisting of infinitely many numbers of concentric circles. If the circle is cut along the radius as shown in the figure and if all those infinitely many lines (circumferences of the concentric circles) are arranged in the form of a right-angled triangle or in the form of an isosceles triangle, then a right-angled triangle or an isosceles triangle is obtained whose base is \(2\pi r\) and height is \(r\).

So, the area of the right-angled triangle or an isosceles triangle \( = \frac{1}{2} \times {\rm{base \times height}} = \frac{1}{2} \times 2\pi r \times r = \pi {r^2}\)

In this case, the area of the circle \( = \) the area of the right-angled triangle or an isosceles triangle \( = \pi {r^2}\)

Circles having the same centre and different radii are called concentric circles.

**Perimeter (Circumference)**

\({C_1}\) and \({C_2}\) be the circumference and \(r\) and \(R\) be the radius of the inner circle and the outer circle, respectively.

Circumference of the outer circle \({C_1} = 2\pi r\)

\( \Rightarrow r = \frac{{{c_1}}}{{2\pi }}\)

Circumference of the inner circle \({C_2} = 2\pi R\)

\( \Rightarrow {\rm{R}} = \frac{{{C_2}}}{{2\pi }}\)

Therefore, the width of the concentric circles \( = (R – r) = \frac{{{c_2}}}{{2\pi }} – \frac{{{c_1}}}{{2\pi }}\)

**Area**

The basic formula for the area of a circle, area \( = \pi {r^2}\) can be applied to find the area enclosed between any two concentric circles.

If \(R\) be the radius of the larger circle and \(r\) be the radius of the smaller circle (both being concentric circles), then the area enclosed between the two concentric circles (shown shaded in the figure) is given by \(\pi {R^2} – \pi {r^2} = \pi \left( {{R^2} – {r^2}} \right)\).

If the circle is divided into two halves, then each part is called a semi-circle.

**Perimeter (Circumference)**

The circumference of the semi-circle equals half of the circumference of the circle.

Circumference of semi-circle \( = \frac{1}{2} \times 2\pi r = \pi r\).

**Area**

The length of the diameter is twice the length of the radius. So, if \(r\) be the radius and \(d\) be the diameter of a circle, then \(d = 2r\) or \(r = \frac{d}{2}\).

Hence, the area of a semi-circle is \( = \frac{1}{2} \times \pi {r^2} = \frac{1}{2} \times \pi \times {(r)^2} = \frac{1}{2} \times \pi \times {\left( {\frac{d}{2}} \right)^2} = \frac{{\pi {d^2}}}{8}\).

**Q.1. A \(15\) inches (diameter) pizza is served to Ram and his friends. Calculate its circumference.**

** Ans:** Diameter of pizza \((d) = 15\)

The formula for the circumference of the circle in terms of diameter \(C = \pi d\)

\(C = \pi \times 15 = 15\pi \) inches

Hence, the circumference of pizza is \(15\pi \).

**Q.2. The wheel of a wheelchair has a diameter of \(56\;\,{\rm{cm}}\). If the wheel rotates once, how much distance does the wheelchair move? (Use: \(\pi = \frac{{22}}{7}\) )**** Ans:** If the wheel rotates once, the wheelchair will move by a distance equal to the circumference of the wheel.

Given, the diameter of the wheel \((d) = 56\;{\rm{cm}}\)

We know that the circumference of a wheel \(C = \pi d\)

\(C = \frac{{22}}{7} \times 56 = 176\;\,{\rm{cm}} = 1\;\,{\rm{m}}\,76\;\,{\rm{cm}}\)

Thus, the wheelchair moves \(1\;\,{\rm{m}}\,76\;\,{\rm{cm}}\) in one revolution of the wheel.

*Q.3. Find the area and perimeter of the circle with a radius \(9\;\,{\rm{m}}\). (Use: \({\pi = \frac{{22}}{7}}\) )*** Ans:** Given: Radius of the circle \((r) = 9\;{\rm{m}}\).

The perimeter of the circle with radius \(‘r’\) is \(2\pi r\).

\( = 2 \times \frac{{22}}{7} \times 9 = 56.57\;{\rm{m}}\)

The area of the circle with radius \(‘r’\) is \(\pi {r^2}\)

\( = \frac{{22}}{7} \times {9^2} = 254.57\;{{\rm{m}}^2}\)

**Q.4. Find the area of the circle, whose diameter is \(4.4\;{\rm{cm}}\) (Use: \({\pi = \frac{{22}}{7}}\) )**** Ans:** Given, the diameter of the circle \((d) = 4.4\;{\rm{cm}}\)

We know that radius of the circle is half the diameter. \(r = \frac{d}{2} = \frac{{4.4}}{2} = 2.2\;{\rm{cm}}\)

The area of the circle with radius \(‘r’\) is \(\pi {r^2}\).

\( = \frac{{22}}{7} \times {2.2^2} = \frac{{106.48}}{7}\;{\rm{c}}{{\rm{m}}^2} = 15.21\;{\rm{c}}{{\rm{m}}^2}\)

Hence, the area of the circle is \(15.21\;{\rm{c}}{{\rm{m}}^2}\)

**Q.5. Given the radius of a circle is \(r\,{\rm{cm}}\) and if it is doubled, then what will be the circumference of the new circle***.*** Ans: **Given, the radius of the circle \( = r\;{\rm{cm}}\)

Then, the circumference of the circle \( = 2\pi r\)

If the radius of the circle is doubled then, the new radius \(R = 2r\;{\rm{cm}}\)

Therefore, the circumference of the new circle \( = 2\pi R = 2\pi \times 2r = 4\pi r\)

Hence, the circumference of the new circle is \(4\pi r\).

In this article, we have learned about the circle and its different components, the definition of the perimeter and area of a circle, the perimeter and area of a circle formula, the perimeter and area of a circle formula using diameter.

The derivation of the area of a circle using the area of a rectangle or using the area of a triangle, the perimeter and area of concentric circles, the perimeter and area of the semi-circle, some interesting frequently asked questions.

We have provided some frequently asked questions here:

**Q.**1.

And, the circumference of the circle \( = 2\pi r\)

In both cases, \(r\) is the radius of the circle.

*Q.2. What are the units of area and perimeter of a circle?*** Ans:** The perimeter or circumference of a circle is generally measured in \({\rm{m}},\,{\rm{cm}},\,{\rm{ft}},\,{\rm{km}}\) and \({\rm{inch}}\). The area of the circle is measured in \({{\rm{m}}^2},\;\,{\rm{c}}{{\rm{m}}^2},\,{\rm{f}}{{\rm{t}}^2}\) and \({\rm{sq}}{\rm{.inches}}\).

*Q.3.*** The circumferences of the two circles are in the ratio**\(1:x\)

So, \(\frac{{{A_1}}}{{{A_2}}} = \frac{{\pi \times r_1^2}}{{\pi \times r_2^2}} = \frac{1}{{{x^2}}}\).

*Q.4.**What is the difference between the area and perimeter of a circle?*** Ans: **The area of a circle is the space occupied by the circle in a two-dimensional plane. The perimeter is the length of the boundary of the circle.

*Q.5. What is the area and perimeter of a circle?*** Ans:** Circumference of a circle or perimeter of a circle is the measure of the length of the boundary of the circle. The area of a circle is defined by the space or region occupied by a circle in a two-dimensional plane.

*We hope this detailed article on Perimeter and Area of Circle helps you in your preparation. If you get stuck do let us know in the comments section below and we will get back to you at the earliest.*