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Area of Circle

Area of Circle

Edited By Team Careers360 | Updated on Jul 02, 2025 05:18 PM IST

Area of circle is the region covered by the circle in a two-dimensional plane. It can be easily calculated using the formula, $\mathrm{A}=\pi \mathrm{r}^2$, (Pi r-squared), $r$ being the term used for radius.

This is an important formula which is useful for calculating the space under a circular field. We have various real life examples giving the usefulness of practical applications of area of a circle. Imagine you need to add a mosquito net for a circular window, to calculate the mosquito net required, the area of circle is important. Like if some guests are arriving at our home, how much area of the table cloth we need in case our table is in the form of a circle. Other few examples are the area of a pizza, area of circular garden, area of a circular stadium, etc. Now we will learn about these concepts in detail.

This Story also Contains
  1. Circle
  2. What is the Area of Circle?
  3. How to Find Area of Circle?
  4. Difference Between Area and Circumference of Circle
  5. Relation Between Area of Square and Area of Circle
  6. Solved Examples based on the Area of Circle
Area of Circle
Area of Circle

In this article we will discuss about the area of circle, area of circle formula, area of circle using other shapes, how to find area of circle, difference between circumference and area of circle and relation between area of square and circle.

Circle

A circle is a closed, round geometric shape. Technically, a circle is a point moving around a fixed point at a fixed distance away from the point. It can be said that a circle is a closed curve where its outer line is equal in distance from the center. The fixed distance from the point is known as the radius of the circle.

Radius: It can be simply defined as the line that joins the center of the circle to the outer boundary or circumference. It is generally represented by ' $r$ ' or ' $R$ '.

Diameter: It is defined as the line that divides the circle into two equal parts or halves and is represented by '$d$' or '$D$'. Hence,

$
d=2 r \text { or } D=2 R
$

And so,

$
r=\frac{d}{2} \text { or } R=\frac{D}{2}
$

Circumference of a Circle

Circumference of Circle is defined as the length of the boundary of the circle. We define the circumference of the circle with the knowledge of a term known as 'pi'.

The perimeter of the circle is equal to the length of its boundary or circumference. If we try wrapping a string around a circle and then unfold it to calculate its length, then upon unfolding it and measuring it we get the circumference of the circle. The formula of circumference is,

Circumference or Perimeter $=2 \pi r$ units, where $r$ is the radius of the circle.
$\pi$, can be pronounced as 'pi', the ratio of the circumference of a circle to its diameter. This ratio is the same for every circle $\frac{22}{7}$ or $3.14$.

What is the Area of Circle?

Any 2D geometrical shape always has its own area. The area of a circle is simply defined as the region covered by its boundary and is calculated using the formula $A=\pi r^2$ measured in square units. It means the space covered by it in xy plane which is also known as 2D plane.

Area of Circle Formula

The area of the circle can be calculated from the radius which can further be taken out from the diameter. But these formulae provide the shortest method to find the area of a circle. Suppose a circle has a radius 'r' then the area of circle $=\pi r^2$. The area of circle with diameter is $\frac{\pi d^2}{4}$ in square units, where $\pi=\frac{22}{7}$ or $3.14$, and $d$ is the diameter.

Area of a circle formula, $A=\pi r^2$ square units
Circumference or Perimeter of the circle$=2 \pi r$ units

Formula of Area of Circle with Radius ($r$)

The area of a circle having radius $r$ can be calculated by using the formulas:
- Area of circle $=\pi \times r^2$, where ' $r$ ' is the radius.

Formula of Area of Circle with Diameter ($d$)

- Area $=(\frac{\pi}{4}) \times d^2$, where ' $d$ ' is the diameter.
- Area $=\frac{C^2}{ 4} \pi$, where ' $C$ ' is the circumference.

The table below summarises all the formulas discussed.

Area of a circle when the radius is known.

$\pi r^2$

Area of a circle when the diameter is known.

$(\frac{\pi}{4}) \times d^2$

Area of a circle when the circumference is known.

$\frac{C^2}{ 4} \pi$

Area of Circle Using other Shapes

Area of a circle can be visualized and proved using two methods, as follows :

  • Determining the circle's area using rectangles

  • Determining the circle's area using triangles

Let us understand both methods one-by-one.

Using Areas of Rectangles

The circle is divided into $16$ equal sectors, and the sectors are arranged as shown in figure. The area of the circle will be equal to that of the parallelogram-shaped figure formed by the sectors cut out from the circle. Since the sectors have equal area, each sector will have an equal arc length. The orange coloured sectors will contribute to half of the circumference, and green coloured sectors will contribute to the other half. If the number of sectors cut from the circle is increased, the parallelogram will finally look like a rectangle with length equal to $\mathrm{\pi r}$ and breadth equal to $r$.


The area of a rectangle (A) will also be the area of a circle. So, we have
- $\mathrm{A}=\pi \times \mathrm{r} \times \mathrm{r}$
- $A=\pi r^2$

Using Area of Triangles

We fill the circle having radius r with concentric circles. After cutting the circle along the indicated line in figure and spreading the lines, the result will be a triangle. The base of the triangle will be equal to the circumference, and its height will be equal to the radius of the circle.


So, the area of the triangle (A) will be equal to the area of the circle. We have

$
\begin{aligned}
& A=\frac{1}{2} \times \text { base } \times \text { height } \\
& A=\frac{1}{2} \times(2 \pi r) \times r \\
& A=\pi r^2
\end{aligned}
$

Surface Area of Circle

A circle is 2-D representation of a sphere. The total area that is taken inside the boundary of the circle is only the surface area of the circle. Hence, we may conclude by saying that both the terms provide us with the same result only. Sometimes, even the volume of a circle is also used to direct towards the area of a circle.

Therefore, the surface area of circle $=A=\pi x r^2$

How to Find Area of Circle?

To find the area of circle we should know the radius or diameter of the circle.
For example, if the radius of the circle is 10 cm, then its area will be:
Area of circle with 10 cm radius $=\pi r^2=\pi(10)^2=\frac{22}{ 7} \times 10 \times 10=314.28 \mathrm{sq} . \mathrm{cm}$.
We can find the area with the help of the following relations:

$
\begin{aligned}
& \mathrm{C}=2 \pi r \\
& \mathrm{r}=\frac{\mathrm{C}}{2 \pi}
\end{aligned}
$

Difference Between Area and Circumference of Circle


Circumference

Area

Definition

The length of the circle's boundary.

The amount of space within the circle.

Units

Same length as the unit. Example: cm, in, ft, etc.

It is measured in square units. Example: cm2, in2, ft2, etc.

Formula

$2 \pi r$

$\pi r^2$

Relationship With Radius

Circumference is directly proportional to the radius.

The area is directly proportional to the square of the radius.

Relationship With Diameter

Circumference is directly proportional to the diameter.

The area is directly proportional to the square of the diameter.

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Relation Between Area of Square and Area of Circle

The area of circle is estimated to be $80 \%$ of area of square, provided that the diameter of the circle and length of side of the square is the same.

For example, if area of square is 100 sq. unit, then the area of circle will be approximately 80 sq.unit of it.

Solved Examples based on the Area of Circle

Example 1: What is the radius of the circle whose surface area is 100 sq.cm?
Solution:

By formula of area of circle, we know that:

$
A=\pi \times r^2
$

Now, substituting the value in the formula for area of circle:

$
\begin{aligned}
& 100=\pi \times r^2 \\
& 100=3.14 \times r^2 \\
& r^2=\frac{100}{3.14} \\
& r^2=31.84 \\
& r=\sqrt{31.34} \\
& r=5.64 \mathrm{~cm}
\end{aligned}
$


Example 2: What is the circumference and the area of circle if the radius is 9 cm .

Solution:
Given: Radius, $r=9 \mathrm{~cm}$.
We know that the circumference or perimeter of the circle is $2 \pi r \mathrm{~cm}$.
Now, substitute the radius value in the formula for circumference of circle,

$
\begin{aligned}
& \mathrm{C}=2 \times(\frac{22}{7}) \times 9 \\
& \mathrm{C}=56.57 \mathrm{~cm}
\end{aligned}
$

Thus, the circumference of the circle is 56.57 cm .
Now, the area of circle is $\pi \mathrm{r}^2 \mathrm{~cm}^2$

$
\begin{aligned}
& A=(\frac{22}{7}) \times 9 \times 9 \\
& A=254.57 \mathrm{~cm}^2
\end{aligned}
$

Example 3: Find the area of circle whose longest chord is 11cm.

Solution:
Given that the longest chord of a circle is 11 cm .
We know that the longest chord of a circle is nothing but the diameter of circle.
Hence, $d=11 \mathrm{~cm}$.
So, $r= \frac{d}{2}=\frac{11}{2}=5.5 \mathrm{~cm}$.
The formula to calculate the area of circle is given by,
$A=\pi r^2$ square units.
Substitute $\mathrm{r}=5.5 \mathrm{~cm}$ in the formula of area of circle, we get
$\begin{gathered}A=\left(\frac{22}{7}\right) \times 5.5 \times 5.5 \mathrm{~cm}^2 \\ A=\left(\frac{22}{7}\right) \times 30.25 \mathrm{~cm}^2 \\ A=\frac{665.5}{7} \mathrm{~cm}^2 \\ A=95.07 \mathrm{~cm}^2\end{gathered}$
Therefore the area of circle is $95.07 \mathrm{~cm}^2$.

Example 4: Rohan and his friends ordered a pizza on Saturday night. Each slice was 5 cm in length.
Calculate the area of the pizza that was ordered by Rohan. You can assume that the length of the pizza slice is equal to the pizza's radius.

Solution:
A pizza is circular in shape. So we can use the area of circle formula to calculate the area of the pizza. Radius is 5 cm .

Area of Circle formula $=\pi r^2=3.14 \times 5 \times 5=78.5$
Area of the Pizza $=78.5 \mathrm{ sq} . \mathrm{cm}$.


Example 5: What is the area of circle with a radius 4 m ?

Solution:
The measure of the radius is given to us which is 4 m . The only thing we have to do is substitute the value of the radius into the formula then simplify.

Therefore $A=3.14 \times 4 \times 4=50.24 \mathrm{~m} \mathrm{sq}$.

List of Topics Related to Area of Circle



Frequently Asked Questions (FAQs)

1. What is area of circle?

The area of circle is defined as the region covered or enclosed by its boundary and is calculated using the formula $A=\pi r^2$ measured in square units.

2. What is the formula of area of circle?

The formula to calculate the area of circle is pi times the radius squared $\left(A=\pi r^2\right)$.

3. How to find area of circle ?

As we know, the area of circle is equal to pitimes square of its radius, i.e. $\pi x \mathrm{r} 2$. To find the area of circle we have to know the radius or diameter of the circle. For example, if the radius of circle is 7 cm , then its area will be: Area of circle with 7 cm radius $=\pi r^2=\pi(7)^2=\frac{22}{7} \times 7 \times 7=22 \times 7=154$ sq. cm.

4. What is the perimeter of circle?

The perimeter of circle is the circumference, which is equal to twice of product of pi $(\pi)$ and radius of circle, i.e., $2 \pi r$.

5. What is the area of circle with radius 2 cm , in terms of $m$ ?

We are given, $r=2 \mathrm{~cm}$.
We know that the area of circle is $\pi r^2$ square units
Hence, $A=\pi \times 2^2=4 \pi \mathrm{~cm}^2$.

6. How does changing the radius affect the area of a circle?
Changing the radius has a significant impact on the area of a circle due to the squared relationship. Doubling the radius will quadruple the area, while tripling the radius will increase the area by a factor of nine. This non-linear relationship is why small changes in radius can lead to large changes in area.
7. Can the area of a circle ever be a whole number?
While it's possible for a circle's area to be very close to a whole number, it's extremely rare for it to be exactly a whole number due to the irrational nature of π. The area will almost always be an irrational number (a number with endless, non-repeating decimal places) because π itself is irrational.
8. How do you find the radius if you know the area of a circle?
To find the radius when you know the area, you need to use the inverse of the area formula. Start with A = πr², then solve for r: r = √(A/π). This means you divide the known area by π and then take the square root of the result to find the radius.
9. What's the relationship between the area of a circle and the area of a square with sides equal to the circle's diameter?
The area of a circle is always smaller than the area of a square with sides equal to the circle's diameter. Specifically, the circle's area is π/4 (approximately 0.7854) times the area of the square. This relationship demonstrates how the circular shape efficiently covers area compared to a square.
10. How does the area of a circle compare to the area of its inscribed square?
The area of a circle is always larger than the area of its inscribed square (a square that fits perfectly inside the circle, touching it at four points). The ratio of the circle's area to the inscribed square's area is π/2 (approximately 1.5708). This shows how much more area the circle covers beyond the largest square that can fit inside it.
11. What's the difference between diameter and radius in calculating circle area?
The diameter is twice the length of the radius. When calculating the area of a circle, we use the radius in the formula A = πr². If you have the diameter (d), you can find the area using A = π(d/2)², where d/2 converts the diameter to radius. Always ensure you're using the radius, not the diameter, in the area formula.
12. How is π (pi) related to the area of a circle?
π (pi) is a fundamental constant in circle geometry, representing the ratio of a circle's circumference to its diameter. In the area formula A = πr², π acts as a scaling factor that, when multiplied by the squared radius, gives the exact area of the circle. π is essential because it captures the unique properties of circular shapes.
13. Why does the area of a circle use r² instead of just r?
The area of a circle uses r² because it represents the two-dimensional space covered by the circle. As the radius increases, the area grows quadratically, not linearly. This squared relationship reflects how the circle expands in both length and width simultaneously as the radius increases.
14. How do you calculate the area of a semicircle?
To calculate the area of a semicircle (half a circle), you use half of the full circle area formula: A = (1/2)πr². This is because a semicircle is exactly half of a full circle. Alternatively, you can calculate the full circle's area and then divide by 2.
15. Can you have a negative area for a circle?
No, the area of a circle cannot be negative. Area represents the amount of two-dimensional space enclosed by the circle's boundary, which is always a positive quantity. Even if you're working with negative coordinates or radii, the area calculation will always yield a positive result due to the squaring of the radius in the formula A = πr².
16. What is the formula for the area of a circle?
The formula for the area of a circle is A = πr², where A is the area, π (pi) is approximately 3.14159, and r is the radius of the circle. This formula represents the relationship between the circle's radius and its total surface area.
17. What's the relationship between the areas of concentric circles?
The difference in area between two concentric circles (circles with the same center but different radii) is always π times the difference of the squares of their radii. If the outer circle has radius R and the inner circle has radius r, the area between them is π(R² - r²). This relationship is useful in many practical applications, like calculating the area of rings or annuli.
18. How do you find the area of a circular ring (annulus)?
To find the area of a circular ring (annulus), calculate the difference between the areas of the outer and inner circles. If R is the radius of the outer circle and r is the radius of the inner circle, the formula is A = π(R² - r²). This can also be written as A = π(R+r)(R-r), which is useful when you know the sum and difference of the radii.
19. How is the area of a circle related to the concept of dimensional analysis?
Dimensional analysis helps verify the correctness of the circle area formula. The radius r has dimension of length (L), and area should have dimension L². In A = πr², π is dimensionless, and r² has dimension L², resulting in the correct dimension for area. This analysis ensures the formula is consistent with the physical meaning of area.
20. What's the difference between exact and approximate values when calculating circle area?
The exact value of a circle's area uses the symbol π in the result, like 9π square units for a circle with radius 3. An approximate value replaces π with a decimal approximation, usually 3.14159 or a rounded version, giving a result like 28.27 square units. Exact values are more precise but less practical for some applications.
21. Why does π appear in the formula for circle area?
π appears in the circle area formula because it's a fundamental constant that relates circular measurements. It represents the ratio of a circle's circumference to its diameter, which is consistent for all circles. In the area formula, π acts as a scaling factor that, when multiplied by r², gives the exact area enclosed by the circle's curved boundary.
22. What's the relationship between the area of a circle and the area of an equilateral triangle with the same perimeter?
A circle always has a larger area than an equilateral triangle with the same perimeter. The circle is the shape that encloses the maximum area for a given perimeter. This property, known as the isoperimetric inequality, demonstrates the circle's efficiency in enclosing area, which is why it's often found in nature and used in engineering designs.
23. What's the difference between arc length and area of a sector?
Arc length is the distance along the curved part of a circle, while the area of a sector is the two-dimensional space enclosed by two radii and an arc. Arc length is a one-dimensional measurement (like circumference), while sector area is two-dimensional (like circle area). They use different formulas and represent different aspects of circular geometry.
24. How does the concept of limits relate to the area of a circle?
The concept of limits is crucial in understanding the area of a circle. As we increase the number of sides in a regular polygon inscribed in a circle, its area approaches the circle's area as a limit. This process, known as exhaustion, was used by ancient mathematicians to approximate π and circle areas, and it forms the foundation for the modern calculus approach to circle area.
25. What's the connection between the area of a circle and the volume of a sphere?
The formula for the volume of a sphere (V = 4/3πr³) is closely related to the circle area formula (A = πr²). The sphere volume can be thought of as the integral of circle areas as they sweep through the sphere's diameter. This connection highlights how 2D circular concepts extend into 3D spherical geometry.
26. How do you estimate the area of an irregular shape using circles?
To estimate the area of an irregular shape using circles, you can use a method called circle packing. Place circles of various sizes within the shape, calculate their total area, and use this as an approximation. Alternatively, you can circumscribe and inscribe circles around the shape to get upper and lower bounds for its area.
27. Why isn't the area of a circle simply πr like the circumference is 2πr?
The area of a circle isn't πr because area represents two-dimensional space, while circumference is a one-dimensional measurement. The r² in the area formula accounts for the circle's expansion in both dimensions as the radius increases. The linear relationship in the circumference formula (2πr) reflects its one-dimensional nature.
28. How does the area of a circle change if you double its circumference?
If you double the circumference of a circle, you're effectively doubling its radius. Since the area depends on the square of the radius (A = πr²), doubling the radius will quadruple the area. This demonstrates the non-linear relationship between a circle's linear measurements (like circumference) and its area.
29. How is the area of a circle related to integration in calculus?
The area of a circle can be derived using calculus through integration. By integrating the function of a semicircle (y = √(r² - x²)) from -r to r, we obtain the area formula πr². This connection demonstrates how calculus can be used to confirm and derive geometric formulas, linking algebraic and geometric concepts.
30. How do you find the area of a circle without using π?
While π is fundamental to circle geometry, you can approximate a circle's area without directly using π by drawing the circle on a grid and counting the squares it covers. This method, known as pixel counting or grid approximation, gives an estimate that improves with finer grids. However, for precise calculations, using π is necessary.
31. How does the area of a circle compare to its circumference squared?
The area of a circle (A = πr²) is always exactly 1/4π times the square of its circumference (C = 2πr). This relationship can be expressed as A = C²/4π. This connection between area and circumference demonstrates how these two fundamental circle measurements are interrelated through π.
32. How do you calculate the area of a circle given its circumference?
To find a circle's area from its circumference, first calculate the radius using the circumference formula C = 2πr, rearranged as r = C/(2π). Then use this radius in the area formula A = πr². Combining these steps gives A = C²/(4π), allowing direct calculation of area from circumference.
33. What's the difference between πr² and 2πr in circle measurements?
πr² represents the area of a circle, while 2πr represents its circumference. The key difference is that area is a two-dimensional measurement (square units), while circumference is one-dimensional (linear units). This is why r is squared in the area formula but not in the circumference formula, reflecting the different dimensions they measure.
34. Why does doubling the radius quadruple the area of a circle?
Doubling the radius quadruples the area because of the squared term in the area formula A = πr². When you double r to 2r, the new area becomes A = π(2r)² = 4πr², which is four times the original area. This quadratic growth demonstrates the non-linear relationship between a circle's radius and its area.
35. What's the significance of the unit circle in understanding circle area?
The unit circle, a circle with radius 1 centered at the origin, is crucial for understanding circle area. Its area is exactly π square units, making it a reference point for all circle calculations. Scaling the unit circle by a factor r gives a circle with radius r and area πr², directly yielding the general area formula.
36. How does the area of a circle change under scaling transformations?
When a circle is scaled by a factor k, its area is multiplied by k². If the original radius is r, the new radius is kr, and the new area is π(kr)² = k²πr², which is k² times the original area. This quadratic scaling demonstrates why small changes in radius can lead to large changes in area.
37. What's the relationship between the area of a circle and the area of its circumscribed square?
The area of a circle is always π/4 (approximately 0.7854) times the area of its circumscribed square (the smallest square that completely contains the circle). This ratio demonstrates how much of the square's area the circle occupies and is useful in comparing circular and square geometries.
38. How do you find the radius of a circle that has twice the area of another circle?
To find the radius of a circle with twice the area of another, use the relationship between their areas: 2πr₁² = πr₂², where r₁ is the radius of the original circle and r₂ is the radius of the new circle. Simplifying, we get r₂ = r₁√2. This shows that to double the area, you need to increase the radius by a factor of √2 (approximately 1.414).
39. What's the importance of significant figures when calculating circle area?
Significant figures are crucial in circle area calculations because of the irrational nature of π. The precision of your result depends on how many decimal places of π you use and the precision of your radius measurement. It's important to consider the level of precision needed for your application and to not report more significant figures than your least precise input value.
40. How does the area of a circle relate to the concept of radians?
Radians and circle area are closely related. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius. The area of a sector with angle θ in radians is given by A = (1/2)r²θ. This formula simplifies to the full circle area πr² when θ = 2π radians, elegantly connecting angular and area measurements.
41. What's the difference between using diameter and radius in circle area calculations?
While the standard area formula uses radius (A = πr²), you can also calculate area using diameter: A = π(d/2)², where d is the diameter. Using diameter requires an extra step (dividing by 2) to convert to radius. It's crucial to identify whether you're given radius or diameter to avoid errors, as using diameter directly in the radius formula would result in an area four times too large.
42. How do you find the area of a circular sector?
The area of a circular sector (a "pie slice" of a circle) is calculated using the formula A = (θ/2π)πr², where θ is the central angle in radians and r is the radius. This can be simplified to A = (1/2)r²θ. The formula represents the fraction of the full circle area corresponding to the fraction of the full angle (2π radians) that the sector occupies.
43. What's the relationship between circle area and the golden ratio?
While the golden ratio (φ ≈ 1.618) isn't directly involved in circle area calculations, it appears in some interesting circle-related geometries. For example, if you inscribe a regular pentagon in a circle, the ratio of the pentagon's area to the circle's area is related to the golden ratio. This connection highlights how fundamental constants like π and φ interrelate in geometry.
44. How do you calculate the area of a circle in terms of its tangent line?
A circle's area can be expressed in terms of the length of its tangent line from a point outside the circle. If t is the length of the tangent line and d is the distance from the point to the circle's center, the area is A = πt²(d²/(d²-t²)). This formula, while less common, demonstrates how circle properties can be expressed through external measurements.
45. What's the connection between circle area and the concept of pi day?
Pi Day, celebrated on March 14 (3/14 in month/day format), honors the mathematical constant π, which is crucial in calculating circle area. The date represents the first three digits of π (3.14). Activities often involve circular objects and area calculations, highlighting the practical applications of π and circle geometry in everyday life and various scientific fields.
46. How does the area of a circle relate to its moment of inertia?
The moment of inertia of a circular disk about its center is directly related to its area. The formula is I = (1/2)mr², where m is the mass and r is the radius. Since the mass can be expressed as density times area (m = ρπr²), the moment of inertia becomes I = (1/2)ρπr⁴. This shows how the area (πr²) influences the disk's rotational properties.
47. What's the significance of circle area in probability and statistics?
Circle area plays a crucial role in probability and statistics, particularly in understanding normal distributions. The "bell curve" of a normal distribution is related to the exponential of a negative quadratic function, which connects to circle geometry. Additionally, circular probability distributions like
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