Area of a Circle – Definition, Formula, Derivation and Examples

Area of a Circle

Area of a circle is the region occupied by the circle in a two-dimensional plane. It can be determined well using a rule, A = πr2, ( Pi r-squared ) where r is the spoke of the circle. The unit of area is the square unit, such as m2, cm2, etc .

Area of Circle = πr 2 or πd 2 /4, square units
where π = 22/7 or 3.14

The area of the circle formula is useful for measuring the space occupied by a circular field or a plot. Suppose you have the plat to fence it, then the area formula will help you to check how much fencing material is required. Or suppose you have to buy a table fabric, then how many portions of fabric is needed to cover it completely .
Hence, the concept of area arsenic well as the perimeter is introduced in Maths, to figure out such scenarios. But, one coarse doubt that arises among most people is “ does a encircle have volume ? ”. The answer is “ No ”. Since a traffic circle is a two-dimensional supreme headquarters allied powers europe, it does not have volume. It has only an sphere and circumference. so, we don ’ t have the book of a circle. In this article, let us discuss in detail the area of a circle, surface area and its circumference with examples .

What is a Circle?

A traffic circle closed plane geometric condition. In technical foul terms, a circle is a locus of a point moving around a fixed point at a repair distance away from the point. Basically, a encircle is a shut swerve with its out line equidistant from the center. The fixate distance from the point is the radius of the circle. In real life, you will get many examples of the r-2 such as a roulette wheel, pizza, a circular ground, etc. now let us learn, what are the terms used in the case of a circle .
Radius
The radius of the circle is the pipeline that joins the center of the r-2 to the out boundary. It is normally represented by ‘ roentgen ’ or ‘ R ’. In the convention for the area and circumference of a encircle, spoke plays an crucial role which you will learn late .
Diameter
The diameter of the traffic circle is the line that divides the circle into two equal parts. In an easy means we can say, it is just the double of the radius of the circle and is represented by ‘ five hundred ’ or ‘ D ’. Therefore ,
d = 2r or D = 2R
If the diameter of the set is known to us, we can calculate the radius of the set, such as ;
r = d/2 or R = D/2
Circumference of Circle
A circumference of close figures is defined as the distance of its boundary. When it comes to circles, the margin is given using a different name. It is called the “ Circumference ” of the r-2. This circumference is the length of the limit of the circle. If we open the circle to form a true occupation, then the duration of the straight line is the circumference. To define the circumference of the circle, cognition of a condition known as ‘ pi ’ is required. Consider the circle shown in the fig. 1, with center at O and radius radius .
Circumference of circle
The circumference of the circle is peer to the duration of its limit. The length of lasso which wraps around its boundary absolutely will be equal to its circumference, which can be measured by using the formula :
Circumference / Perimeter = 2πr units
where radius is the radius of the circle .
π, read as ‘ pi ’ is defined as the ratio of the circumference of a circle to its diameter. This ratio is the like for every circle. Consider a r-2 with spoke ‘ r ’ and circumference ‘ C ’. For this set

  • π = Circumference/Diameter
  • π= C/2r
  • C = 2πr

The lapp is shown in figure. 2 .
perimeter of circle

What is Area of Circle?

Area of a circle is the area covered or enclosed within its limit. It is measured in squarely units .
Any geometric shape has its own area. This area is the region that occupies the form in a planar plane. now we will learn about the area of the encircle. So the area covered by one complete bicycle of the spoke of the circle on a planar plane is the area of that r-2. now how can we calculate the area for any round object or space ? In this case, we use the rule for the circle ’ randomness area. Let us discuss the formula now.

Area of a Circle Formula

Let us take a circle with radius roentgen .
Area of a circle
In the above calculate, we can see a circle, where radius gas constant from the center ‘ o ’ to the boundary of the r-2. then the area for this r-2, A, is adequate to the intersection of private detective and square of the spoke. It is given by ;  

Area of a Circle, A = πr2 square units

here, the value of private detective, π = 22/7 or 3.14 and radius is the radius.

Derivation of Area of Circle

Area of a circle can be visualized & proved using two methods, namely

  • Determining the circle’s area using rectangles
  • Determining the circle’s area using triangles

Let us understand both the methods one-by-one.

Using Areas of Rectangles

The circle is divided into 16 equal sectors, and the sectors are arranged as shown in libyan islamic fighting group. 3. The area of the circle will be equal to that of the parallelogram-shaped calculate formed by the sectors cut out from the circle. Since the sectors have equal area, each sector will have an equal discharge length. The red bleached sectors will contribute to one-half of the circumference, and blue coloured sectors will contribute to the other half. If the number of sectors cut from the lap is increased, the parallelogram will finally look like a rectangle with length adequate to πr and breadth equal to r .
Area of a Circle using Rectangle
The area of a rectangle ( A ) will besides be the area of a circle. sol, we have

  • A = π×r×r
  • A= πr2

Using Areas of  Triangles

Fill the set with radius gas constant with concentric circles. After cutting the set along the indicate line in figure. 4 and spreading the lines, the result will be a triangle. The base of the triangle will be adequate to the circumference of the traffic circle, and its height will be equal to the radius of the circle .
Area of a Circle using Triangles
so, the area of the triangle ( A ) will be equal to the area of the r-2. We have
A = 1/2×base×height
A = 1/2× ( 2πr ) ×r
A = πr2

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Surface Area of Circle

A circle is nothing but the 2-D representation of a sphere. The sum area that is taken inside the limit of the circle is the surface area of the encircle. When we say we want the area of the circle, then we mean the surface area of the circle itself. sometimes, the bulk of a circle besides defines the sphere of a circle .
When the length of the radius or diameter or even the circumference of the circle is already given, then we can use the surface formula to find out the come on area. The surface is represented in square units .
The airfoil area of the traffic circle = A = π x r2

How to Find Area of a Circle?

As we know, the area of traffic circle is adequate to pi times square of its radius, i.e. π x r2. To find the area of circle we have to know the radius or diameter of the circle.
For case, if the spoke of encircle is 7cm, then its area will be :
Area of encircle with 7 cm radius = πr2 = π ( 7 ) 2 = 22/7 x 7 ten 7 = 22 ten 7 = 154 sq.cm .
besides, if we know the circumference of the set, then we can find the area of circle.
How ?
Since, the circumference is 2 times of product of protease inhibitor and radius of circle, such as :
C = 2πr
Therefore, here we can find the value of radius,
roentgen = C/2π
once, we have evaluated the rate of radius, we can well find the area.

Difference Between Square Area and Circle Area

The area of a lap is estimated to be 80 % of area of squarely, when the diameter of the circle and length of side of the square is the like .
Students can besides do an action by inserting a round object into a square human body with lapp diameter and side-length, respectively .
If sphere of squarely is 100 sq.unit, then the area of traffic circle will be approximately 80 sq.unit of it.

Video Lesson

Area of a Circle

Area of a Circle

Solved Examples on Area of a Circle

We have discussed cashbox now the unlike parameters of the circle such as area, circumference or circumference, radius and diameter. Let us solve some problems based on these formulas to understand the concept of area and perimeter in a better way .
Example 1:
What is the radius of the r-2 whose come on area is 314.159 sq.cm ?
Solution:
By the rule of the coat area of the encircle, we know ;
A = π x r2
immediately, substituting the value :
314.159 = π x r2
314.159 = 3.14 x r2
r2 = 314.159/3.14
r2 = 100.05
roentgen = √100.05
r = 10 centimeter
Example 2:
Find the circumference and the area of encircle if the radius is 7 curium .
Solution:
Given : Radius, roentgen = 7 centimeter
We know that the circumference/ circumference of the circle is 2πr curium .
now, substitute the radius respect, we get
C = 2 × ( 22/7 ) × 7
C = 2×22
C = 44 centimeter
frankincense, the circumference of the circle is 44 centimeter .
now, the area of the circle is πr2 cm2
A = ( 22/7 ) × 7 × 7
A = 22 × 7
A = 154 cm2
Example 3:
If the longest chord of a set is 12 centimeter, then find the area of circle .
Solution:
Given that the longest harmonize of a circle is 12 centimeter .
We know that the longest chord of a circle is the diameter .
Hence, d = 12 centimeter .
so, r = d/2 = 12/2 = 6 curium .
The formula to calculate the area of lap is given by ,
A = πr2 square units .
now, substitute roentgen = 6 curium in the convention, we get
A = ( 22/7 ) ×6×6 cm2
A = ( 22/7 ) ×36 cm2
A = 792/7 cm2
A = 113.14 cm2 ( Rouned to 2 decimal fraction places )
consequently the area of circle is 113.14 cm2.

frequently Asked Questions on Area of Circle

What is meant by area of circle?

The area of circle is the region occupied by r-2 in the planar space .

How to calculate the area of a circle?

The area of circle can be calculated by using the recipe :
Area = π x r2, in terms of radius ‘ radius ’.
Area = ( π/4 ) x d2, in terms of diameter, ‘ d ’.
Area = C2/4π, in terms of circumference, ‘ C ’ .

What is the perimeter of circle?

The margin of set is nothing but the circumference, which is adequate to twice of intersection of pi ( π ) and radius of r-2, i, 2πr .

What is the area of a circle with radius 3 cm, in terms of π?

Given, r = 3 centimeter.
We know that the sphere of circle is πr2 square units
Hence, A = π x 32 = 9π cm2 .

Find the circumference of circle in terms of π, whose radius is 14 cm.

We know that the circumference of a set is 2πr units.
Hence, C = 2π ( 14 ) = 28π centimeter .

Find the radius of the circle, if its area is 340 square centimeters.

We know that, Area of a circle = πr2 square units
Hence, 340 =3.14 r2
Hence, r2 = 340/3.14
r2 = 108.28
Hence, radius = 10.4 centimeter.
Hence, spoke of a lap = 10.4 curium

Determine the area of the circle in terms of pi, if radius = 6 cm.

We know that, Area = πr2
A = π ( 6 ) 2
A = 36π
Hence, the area of a encircle is 36π, if the radius is 6 centimeter.

Find the area of a circle, if its circumference is 128 inches.

The area of a lap is 1303.8 square inches if its circumference is 128 inches .
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