Area of a Circle

The area of a circle is the space occupied by the circle in a two-dimensional plane. Alternatively, the space occupied within the boundary/circumference of a circle is called the area of the circle. The formula for the area of a circle is A = πr², where r is the radius of the circle. The unit of area is the square unit, for example, m², cm², in², etc. Area of Circle = πr² or πd²/4 in square units, where (Pi) π = 22/7 or 3.14. Pi (π) is the ratio of circumference to diameter of any circle. It is a special mathematical constant.

The area of a circle formula is useful for measuring the region occupied by a circular field or a plot. Suppose, if you have a circular table, then the area formula will help us to know how much cloth is needed to cover it completely. The area formula will also help us to know the boundary length i.e., the circumference of the circle. Does a circle have volume? No, a circle doesn’t have a volume. A circle is a two-dimensional shape, it does not have volume. A circle only has an area and perimeter/circumference. Let us learn in detail about the area of a circle, surface area, and its circumference with examples.

Area of Circle = πr² or πd²/4, square units

where π = 22/7 or 3.14

A circle is a collection of points that are at a fixed distance from the center of the circle. A circle is a closed geometric shape. We see circles in everyday life such as a wheel, pizzas, a circular ground, etc. The measure of the space or region enclosed inside the circle is known as the area of the circle.

Radius: The distance from the center to a point on the boundary is called the radius of a circle. It is represented by the letter ‘r’ or ‘R’. Radius plays an important role in the formula for the area and circumference of a circle, which we will learn later.

Diameter: A line that passes through the center and its endpoints lie on the circle is called the diameter of a circle. It is represented by the letter ‘d’ or ‘D’.

Diameter formula: The diameter formula of a circle is twice its radius. Diameter = 2 × Radius

d = 2r or D = 2R

If the diameter of a circle is known, its radius can be calculated as:

r = d/2 or R = D/2

Circumference: The circumference of the circle is equal to the length of its boundary. This means that the perimeter of a circle is equal to its circumference. The length of the rope that wraps around the circle’s boundary perfectly will be equal to its circumference. The below-given figure helps you visualize the same. The circumference can be measured by using the given formula:

where ‘r’ is the radius of the circle and π is the mathematical constant whose value is approximated to 3.14 or 22/7. The circumference of a circle can be used to find the area of that circle.

For a circle with radius ‘r’ and circumference ‘C’:

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

Let us understand the different parts of a circle using the following real-life example.

Consider a circular-shaped park as shown in the figure below. We can identify the various parts of a circle with the help of the figure and table given below.

In a Circle	In our park	Named by the letter
Centre	Fountain	F
Circumference	Boundary
Chord	Play area entrance	PQ
Radius	Distance from the fountain to the Entrance gate	FA
Diameter	Straight Line Distance between Entrance Gate and Exit Gate through the fountain	AFB
Minor segment	The smaller area of the park, which is shown as the Play area
Major segment	The bigger area of the park, other than the Play area
Interior part of the circle	The green area of the whole park
Exterior part of the circle	The area outside the boundary of the park
Arc	Any curved part on the circumference.

The area of a circle is the amount of space enclosed within the boundary of a circle. The region within the boundary of the circle is the area occupied by the circle. It may also be referred to as the total number of square units inside that circle.

The area of a circle can be calculated in intermediate steps from the diameter, and the circumference of a circle. From the diameter and the circumference, we can find the radius and then find the area of a circle. 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 = πr² or πd²/4 in square units, where π = 22/7 or 3.14, and d is the diameter.

Area of a circle, A = πr² square units

Circumference / Perimeter = 2πr units

Area of a circle can be calculated by using the formulas:

• Area = π × r², where ‘r’ is the radius.
• Area = (π/4) × d², where ‘d’ is the diameter.
• Area = C²/4π, where ‘C’ is the circumference.

Examples using Area of Circle Formula

Let us consider the following illustrations based on the area of circle formula.

Example 1: If the length of the radius of a circle is 4 units. Calculate its area.

Solution:
Radius(r) = 4 units(given)
Using the formula for the circle’s area,
Area of a Circle = πr²
Put the values,
A = π4²
A =π × 16
A = 16π ≈ 50.27

Answer: The area of the circle is 50.27 squared units.

Example 2: The length of the largest chord of a circle is 12 units. Find the area of the circle.

Solution:
Diameter(d) = 12 units(given)
Using the formula for the circle’s area,
Area of a Circle = (π/4)×d²
Put the values,
A = (π/4) × 12²
A = (π/4) × 144
A = 36π ≈ 113.1

Answer: The area of the circle is 113.1 square units.

Area of a Circle Using Diameter

The area of the circle formula in terms of the diameter is: Area of a Circle = πd²/4. Here ‘d’ is the diameter of the circle. The diameter of the circle is twice the radius of the circle. d = 2r. Generally from the diameter, we need to first find the radius of the circle and then find the area of the circle. With this formula, we can directly find the area of the circle, from the measure of the diameter of the circle.

Area of a Circle Using Circumference

The area of a circle formula in terms of the circumference is given by the formula $\frac{(Circumference{)}^2}{4\pi }$

. There are two simple steps to find the area of a circle from the given circumference of a circle. The circumference of a circle is first used to find the radius of the circle. This radius is further helpful to find the area of a circle. But in this formulae, we will be able to directly find the area of a circle from the circumference of the circle.

Area of a Circle-Calculation

The area of the circle can be conveniently calculated either from the radius, diameter, or circumference of the circle. The constant used in the calculation of the area of a circle is pi, and it has a fractional numeric value of 22/7 or a decimal value of 3.14. Any of the values of pi can be used based on the requirement and the need of the equations. The below table shows the list of formulae if we know the radius, the diameter, or the circumference of a circle.

Area of a circle when the radius is known.	πr2
Area of a circle when the diameter is known.	πd2/4
Area of a circle when the circumference is known.	C2/4π

Why is the area of the circle is πr²? To understand this, let’s first understand how the formula for the area of a circle is derived.

Observe the above figure carefully, if we split up the circle into smaller sections and arrange them systematically it forms a shape of a parallelogram. When the circle is divided into even smaller sectors, it gradually becomes the shape of a rectangle. The more the number of sections it has more it tends to have a shape of a rectangle as shown above.

The area of a rectangle is = length × breadth

The breadth of a rectangle = radius of a circle (r)

When we compare the length of a rectangle and the circumference of a circle we can see that the length is = ½ the circumference of a circle

Area of circle = Area of rectangle formed = ½ (2πr) × r

Therefore, the area of the circle is πr², where r, is the radius of the circle and the value of π is 22/7 or 3.14.

The surface area of a circle is the same as the area of a circle. In fact, when we say the area of a circle, we mean nothing but its total surface area. Surface area is the area occupied by the surface of a 3-D shape. The surface of a sphere will be spherical in shape but a circle is a simple plane 2-dimensional shape.

If the length of the radius or diameter or even the circumference of the circle is given, then we can find out the surface area. It is represented in square units. The surface area of circle formula = πr² where ‘r’ is the radius of the circle and the value of π is approximately 3.14 or 22/7.

Ron and his friends ordered a pizza on Friday night. Each slice was 15 cm in length.

Calculate the area of the pizza that was ordered by Ron. 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 a circle formula to calculate the area of the pizza.

Radius is 15 cm

Area of Circle formula = πr² = 3.14 × 15 × 15 = 706.5

Area of the Pizza = 706.5 sq. cm.

Area of Circle Examples

Example 2: The ratio of the area of 2 circles is 4:9. With the help of the area of circle formula find the ratio of their radii.

Solution:

Let us assume the following:

The radius of the 1st circle = R1

Area of the 1st circle = A1

The radius of the 2nd circle = R2

Area of the 2nd circle = A2

It is given that A1:A2 = 4:9

Example 5: The time shown in a circular clock is 3:00 pm. The length of the minute hand is 21 units. Find the distance traveled by the tip of the minute hand when the time is 3:30 pm.

Solution:

When the minute hand is at 3:30 pm, it covers half of the circle. So, the distance traveled by the minute hand is actually half of the circumference. Distance $=\pi$

(where r is the length of the minute hand). Hence the distance covered = 22/7 × 21 = 22 × 3 = 66 units. Therefore, the distance traveled is 66 units.

Area of a Circle Calculator

How to calculate the area of a circle? Area of a circle formula

Diameter = 2 × Radius.

Area of a circle radius. The radius of a circle calculator uses the following area of a circle formula:

Area of a circle = π × r².

Area of a circle diameter. The diameter of a circle calculator uses the following equation:

Area of a circle = π × (d/2)²,

where:

• π is approximately equal to 3.14. It doesn’t matter whether you want to find the area of a circle using diameter or radius – you’ll need to use this constant in almost every case.

• A sector of a circle is a section of a circle between two radii. You can think of it as a giant slice of pizza.
• If you want to know how to draw a circle?, the equation for the coordinates and the center of a circle in the coordinate system might come in handy.

How to use the area of a circle calculator? Diameter to area and radius to area.

You can easily calculate everything, the area of a circle, its diameter, and its radius, using our area of a circle calculator in a blink of an eye:

1. Determine whether your given value is a diameter or a radius using the picture on the right and definitions available in the section above (you can calculate the area of a circle using its diameter as well as radius).

2. Enter your value into the proper field of the calculator.

3. It didn’t take long – your results are here! We decided to include the step-by-step solution and all the most important data right below the calculator.

Formula

The surface area of a circle is given by the following formula:

A = πr²

where r is the radius of the given circle.

or

A = C²/4π

where C is the circumference of the given circle.

Sample Problems

Question 1. Find the area of a circle given its radius is 8 m.

Solution:

Given: r = 8 m

Since area of a circle = πr²

A = π(8)²

= 64π

= 200.96 m²

Question 2. Find the area of a circle given its circumference is 12 cm.

Solution:

Given: C = 12 cm

Since, A = C²/4π

= 12²/4π

= 11.46 cm²

Question 3. Find the area of a circle given its diameter is 12 cm.

Solution:

Given: D = 12 cm

or, Radius = r = 12/2 = 6 cm

Since A = πr²

= π(6)2

= 113.04 cm²

Question 4. Find the area of a circle given its radius is 9 cm.

Solution:

Given: r = 9 m

Since area of a circle = πr²

A = π(9)²

= 81π

= 254.34 cm²

Question 5. Find the area of a circle given its diameter is 10 cm.

Solution:

Given: D = 10 cm

or, Radius = r = 10/2 = 5 cm

Since A = πr²

= π(5)²

= 78.5 cm²

The Area of a Circle Formula

Apply the Surface Area Formula

Formula for Area From Diameter

Formula for Area From Circumference

What is a Circle?

A circle closed plane geometric shape. In technical terms, a circle is a locus of a point moving around a fixed point at a fixed distance away from the point. Basically, a circle is a closed curve with its outer line equidistant from the center. The fixed distance from the point is the radius of the circle. In real life, you will get many examples of the circle such as a wheel, pizzas, 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 line that joins the center of the circle to the outer boundary. It is usually represented by ‘r’ or ‘R’. In the formula for the area and circumference of a circle, radius plays an important role which you will learn later.

Diameter

The diameter of the circle is the line that divides the circle into two equal parts. In an easy way we can say, it is just the double of the radius of the circle and is represented by ‘d’ or ‘D’. Therefore,

d = 2r or D = 2R

If the diameter of the circle is known to us, we can calculate the radius of the circle, such as;

r = d/2 or R = D/2

Circumference of Circle

A perimeter of closed figures is defined as the length of its boundary. When it comes to circles, the perimeter is given using a different name. It is called the “Circumference” of the circle. This circumference is the length of the boundary of the circle. If we open the circle to form a straight line, then the length of the straight line is the circumference. To define the circumference of the circle, knowledge of a term known as ‘pi’ is required. Consider the circle shown in the fig. 1, with centre at O and radius r.

The perimeter of the circle is equal to the length of its boundary. The length of rope which wraps around its boundary perfectly will be equal to its circumference, which can be measured by using the formula:

Circumference / Perimeter = 2πr units

where r 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 same for every circle. Consider a circle with radius ‘r’ and circumference ‘C’. For this circle

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

The same is shown in fig. 2.

What is Area of Circle?

Area of a circle is the region covered or enclosed within its boundary. It is measured in square units.

Any geometrical shape has its own area. This area is the region that occupies the shape in a two-dimensional plane. Now we will learn about the area of the circle. So the area covered by one complete cycle of the radius of the circle on a two-dimensional plane is the area of that circle. Now how can we calculate the area for any circular object or space? In this case, we use the formula for the circle’s area. Let us discuss the formula now.

Area of a Circle Formula

Let us take a circle with radius r.

In the above figure, we can see a circle, where radius r from the center ‘o’ to the boundary of the circle. Then the area for this circle, A, is equal to the product of pi and square of the radius. It is given by; 

Here, the value of pi, π = 22/7 or 3.14 and r 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 fig. 3. 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 red coloured sectors will contribute to half of the circumference, and blue coloured sectors will contribute to the other half. If the number of sectors cut from the circle is increased, the parallelogram will eventually look like a rectangle with length equal to πr and breadth equal to r.

The area of a rectangle (A) will also be the area of a circle. So, we have

• A = π×r×r
• A = πr²

Using Areas of  Triangles

Fill the circle with radius r with concentric circles. After cutting the circle along the indicated line in fig. 4 and spreading the lines, the result will be a triangle. The base of the triangle will be equal to the circumference of the circle, 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

A = 1/2×base×height

A = 1/2×(2πr)×r

A = πr²

Surface Area of Circle

A circle is nothing but the 2-D representation of a sphere. The total area that is taken inside the boundary of the circle is the surface area of the circle. When we say we want the area of the circle, then we mean the surface area of the circle itself. Sometimes, the volume of a circle also defines the area 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 surface area. The surface is represented in square units.

The surface area of the circle = A =  π x r²

How to Find Area of a Circle?

As we know, the area of circle is equal 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 example, if the radius of circle is 7cm, then its area will be:
Area of circle with 7 cm radius = πr² = π(7)² = 22/7 x 7 x 7 = 22 x 7 = 154 sq.cm.

Also, if we know the circumference of the circle, then we can find the area of circle.
How?
Since, the circumference is 2 times of product of pi and radius of circle, such as:
C = 2πr
Therefore, here we can find the value of radius,
r = C/2π
Once, we have evaluated the value of radius, we can easily find the area.

Difference Between Square Area and Circle Area

The area of a circle is estimated to be 80% of area of square, when the diameter of the circle and length of side of the square is the same.

Students can also do an activity by inserting a circular object into a square shape with same diameter and side-length, respectively.

Solved Examples on Area of a Circle

We have discussed till now the different parameters of the circle such as area, perimeter 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 circle whose surface area is 314.159 sq.cm?

Solution:

By the formula of the surface area of the circle, we know;

A = π x r²

Now, substituting the value:

314.159 = π x r²

314.159 = 3.14 x r²

r² = 314.159/3.14

r² = 100.05

r = √100.05

r = 10 cm

Example 2:

Find the circumference and the area of circle if the radius is 7 cm.

Solution:

Given: Radius, r = 7 cm

We know that the circumference/ perimeter of the circle is 2πr cm.

Now, substitute the radius value, we get

C = 2  × (22/7)× 7

C = 2×22

C = 44 cm

Thus, the circumference of the circle is 44 cm.

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 circle is 12 cm, then find the area of circle.

Solution:

Given that the longest chord of a circle is 12 cm.

We know that the longest chord of a circle is the diameter.

Hence, d = 12 cm.

So, r = d/2 = 12/2 = 6 cm.

The formula to calculate the area of circle is given by,

A = πr² square units.

Now, substitute r = 6 cm in the formula, 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 places)

Therefore the area of circle is 113.14 cm².

Frequently Asked Questions on Area of Circle

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

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

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, r = 10.4 cm.
Hence, radius of a circle = 10.4 cm

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 circle is 36π, if the radius is 6 cm.

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

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

Surface Area of Circle Formula

Solved example

Question: What is the radius of the circle whose surface area is 314.159

GIA SƯ TOÁN BẰNG TIẾNG ANH

GIA SƯ DẠY SAT

Math Formulas

Mọi chi tiết liên hệ với chúng tôi :
TRUNG TÂM GIA SƯ TÂM TÀI ĐỨC
Các số điện thoại tư vấn cho Phụ Huynh :
Điện Thoại : 091 62 65 673 hoặc 01634 136 810
Các số điện thoại tư vấn cho Gia sư :
Điện thoại : 0902 968 024 hoặc 0908 290 601