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## Geometry of a Circle

### Definition

The distance around the boundary of a circle is called the *circumference*.

The distance across a circle through the centre is called the *diameter*.

The distance from the centre of a circle to any point on the boundary is called the *radius*. The radius is half of the diameter; 2r=d.

The line segment that joins two points on the circle is a *chord*. Every diameter is a chord, but not every chord is a diameter.

The area that a chord cuts off is called a *segment*.

The area inside a circle and bounded by two radii is a *sector*.

The length between two points around the circumference of a circle is an *arc*.

#### Circumference

Definition

The formula for calculating the circumference is

C=πd or C=2πr

where d is the diameter and r is the radius.

**Worked Examples**

**Example 1**

The radius of a given circle is r=4cm. Calculate the circumference.

**Solution**

**Example 2**

Find the diameter of a circle with circumference 18cm.

**Solution**

Divide both sides by π:

### Area

**Definition**

The area of a circle with radius r is

**Worked Examples**

Example 1

The radius of a given circle is 2.52.5cm. Find the area of the circle.

Solution

Example 2

The area of a circle is 5050cm². Find the radius.

Solution

Divide both sides by π.

## Circumference of a Circle

In Mathematics, the circumference of any shape defines the path or the boundary that surrounds the shape. In other words, the circumference is also called the perimeter, which helps to identify the length of the outline of any shape. As we know, the **perimeter and area of circle **are the two important parameters of a circle. In this article, we will discuss the “**Circumference of a circle**” or “**Perimeter of circle**” with its definition, formula, methods to find the circle’s circumference with many solved examples.

## Circle’s Circumference

**Circumference of the circle or perimeter of the circle** is the measurement of the boundary of the circle. Whereas the area of circle defines the region occupied by it. If we open a circle and make a straight line out of it, then its length is the circumference. It is usually measured in units, such as cm or unit m.

When we use the formula to calculate the circumference of the circle, then the radius of the circle is taken into account. Hence, we need to know the value of the radius or the diameter to evaluate the perimeter of the circle.

## Circumference of a Circle Formula

The **Circumference (or) perimeter of circle = 2πR**

where,

R is the radius of the circle

π is the mathematical constant with an approximate (up to two decimal points) value of 3.14

Again,

Pi (π) is a special mathematical constant; it is the ratio of circumference to diameter of any circle.

where** C = π D**

C is the circumference of the circle

D is the diameter of the circle

**For example:** If the radius of the circle is 4cm then find its circumference.

Given: Radius = 4cm

Circumference = 2πr

= 2 x 3.14 x 4

= 25.12 cm

### Area of a Circle Formula

Area of any circle is the region enclosed by the circle itself or the space covered by the circle. The formula to find the area of the circle is;

**A = πr ^{2}**

Where r is the radius of the circle, this formula is applicable to all the circles with different radii.

## Perimeter of Semi-Circle

The semi-circle is formed when we divide the circle into two equal parts. Therefore, the perimeter of the semi-circle also becomes half.

Hence, **Perimeter = πr +2r**

## Area of Semi-Circle

Area of the semi-circle is the region occupied by a semi-circle in a 2D plane. The area of the semi-circle is equal to half of the area of a circle, whose radii are equal.

Therefore, **Area = πr ^{2}/2**

Thus, we can define three different formulas to find the perimeter of circle (i.e. circumference of a circle).

**Formula 1:** When the radius of a circle is known.

Circumference of a circle = 2πr

**Formula 2: **When the diameter of a circle is known.

Circumference = πd

**Formula 3: **When the area of a circle is known, we can write the formula to find the perimeter of the circle as:

C = √(4πA)

Here,

C = Circumference of the circle

A = Area of the circle

**Summary**

Circumference of Circle | 2πr |

Area of circle | πr^{2} |

Perimeter of semi-circle | πr + 2r |

Area of semi-circle | πr^{2}/2 |

### Radius of a Circle

The distance from the centre to the outer line of the circle is called a radius. It is the most important quantity of the circle based on which formulas for the area and circumference of the circle are derived. Twice the radius of a circle is called the diameter of the circle. The diameter cuts the circle into two equal parts, which is called a semi-circle.

## What is the Circumference of Circle?

The meaning of circumference is the distance around a circle or any curved geometrical shape. It is the one-dimensional linear measurement of the boundary across any two-dimensional circular surface. It follows the same principle behind finding the perimeter of any polygon, which is why calculating the circumference of a circle is also known as the **perimeter of a circle.**

A circle is defined as a shape with all the points are equidistant from a point at the centre. The circle depicted below has its centre lies at point A.

The **value of pi is approximately 3.1415926535897**… and we use a Greek letter π (pronounced as Pi) to describe this number. The value** π **is a non-terminating value.

For circle A (as given below), the circumference and the diameter will be-

In other words, the distance surrounding a circle is known as the circumference of the circle. The diameter is the distance across a circle through the centre, and it touches the two points of the circle perimeter. **π **shows the ratio of the perimeter of a circle to the diameter. Therefore, when you divide the circumference by the diameter for any circle, you obtain a value close enough to π. This relationship can be explained by the formula mentioned below.

**C/d = π**

Where C indicates circumference and d indicates diameter. A different way to put up this formula is C = π × d. This formula is mostly used when the diameter is mentioned, and the perimeter of a circle needs to be calculated.

**Circumference to Diameter**

We know that the diameter of a circle is twice the radius. The proportion between the circumference of a circle and its diameter is equal to the value of Pi(π). Hence, we say that this proportion is the definition of the constant π.

(i.e) C= 2πr

C= πd (As, d = 2r)

If we divide both sides by the diameter of the circle, we will get the value that is approximately close to the value of π.

Thus, C/d = π.

## How to Find Circumference?

**Method 1: **Since it is a curved surface, we can’t physically measure the length of a circle using a scale or ruler. But this can be done for polygons like squares, triangles and rectangles. Instead, we can measure the circumference of a circle using a thread. Trace the path of the circle using the thread and mark the points on the thread. This length can be measured using a normal ruler.

**Method 2: **An accurate way of knowing the circumference of a circle is to calculate it. For this, the radius of the circle has to be known. The radius of a circle is the distance from the centre of the circle and any point on the circle itself. The figure below shows a circle with radius R and centre O. The diameter is twice the radius of the circle.

### Circumference of a Circle Definition

The circumference of a circle refers to the measure of its boundary. If we open a circle and measure the boundary just like we measure a straight line, we get the circumference of the circle in terms of units of length like centimeters, meters, or kilometers.

Now let us learn about the elements that make up circumference. These are the three most important elements of a circle.

**Center:**The center of the circle is a point that is at a fixed distance from any other point from the circumference.**Diameter:**The diameter is the distance across the circle through the center, it is a line that meets the circumference at both ends and it needs to pass through the center.**Radius:**The radius of a circle is the distance from the center of a circle to any point on the circumference of the circle.

## Circumference of Circle Formula

The formula for the circumference of a circle is expressed using the radius ‘r’ of the circle and the value of ‘pi’. It is expressed as, Circumference of a circle formula = 2πr. While using this circumference formula, if we do not have the value of the radius, we can find it using the diameter. That is, if the diameter is known, it can be divided by 2 to obtain the value of the radius because of the diameter of a circle = 2 × radius. Another way to calculate the circumference of a circle is by using the formula: Circumference = π × Diameter. If we need to calculate the radius or diameter, when the circumference of a circle is given, we use the formula: Radius = Circumference/2π

## How to Find the Circumference of Circle?

Although the circumference of a circle is the length of its boundary, it cannot be calculated with the help of a ruler (scale) like it is usually done for other polygons. This is because a circle is a curved figure. Therefore, to calculate the circumference of a circle, we apply a formula that uses the **radius** or the **diameter** of the circle and the **value of Pi **(π).

Pi is a special mathematical constant with a value approximated to 3.14159 or π = 22/7. The value of π = 22/7 is used in various formulas. It is the ratio of circumference to diameter, where C = πD. Let us consider a practical illustration to understand how to calculate the circumference of a circle with the help of the circumference formula.

**Example:** If the radius of the circle is 25 units, find the circumference of the circle. (Take π = 3.14)

**Solution:** Given, radius = 25 units

Let us write the circumference formula and then we will substitute the value of r (radius) in it.

Circumference of circle formula = 2πr

C = 2 × π × 25

C = 2 × 3.14 × 25 = 157 units

Therefore, the circumference of a circle is 157 units.

**Important Notes on Circumference of a Circle**

- π(Pi) is a mathematical constant that is the ratio of the circumference of a circle to its diameter. It is approximated to π = 22/7or 3.14
- If the radius of a circle is extended further and touches the boundary of the circle, it becomes the diameter of a circle. Therefore, Diameter = 2 × Radius
- The circumference is the distance around a circle or the length of a circle.
- We can find the circumference of a circle using the radius or diameter.
- Circumference formula = π× Diameter; Circumference = 2πr.

## Solved Examples on Perimeter of Circle

**Example 1: **

What is the circumference of the circle with diameter 4 cm?

**Solution:**

Since the diameter is known to us, we can calculate the radius of the circle,

Therefore, Circumference of the Circle = 2 x 3.14 x 2 = 12.56 cm.

**Example**** 2:**

Find the radius of the circle having C = 50 cm.

**Solution: **

Circumference = 50 cm

As per formula, C = 2 π r

This implies, 50 = 2 π r

50/2 = 2 π r/2

25 = π r

or r = 25/π

Therefore, the radius of the circle is 25/π cm.

**Example 3:**

Find the perimeter of circle whose radius is 3 cm?

**Solution:**

Given: Radius = 3 cm.

We know that the circumference or the perimeter of a circle is 2πr units.

Now, substitute the radius value in the formula, we get

C = (2)(22/7)(3) cm

C = 18.857 cm

Therefore, the circumference of circle is 18.857 cm.

**Example 4: **

Calculate the perimeter of circle in terms of π, whose diameter is 10m.

**Solution:**

Given: Diameter = 10m.

Hence, radius = diameter/2 = 10/2 = 5 m.

We know that, perimeter of circle = 2πr units

C = 2π(5) = 10π m.

Therefore, the perimeter of circle in terms of π, whose diameter 10 cm is 10π m.

**Example 5:**

Find the perimeter and area of circle whose radius is 5 cm. [Note: π = 3.14]

**Solution:**

To find: Perimeter and area of circle.

Given that, Radius, r = 5 cm and π = 3.14

As we know, the circumference (or) perimeter of a circle = 2πr units

The area of a circle = πr^{2} square units.

Now, substitute the values in the perimeter and area of circle formula, we get

The area of circle = πr^{2} = 3.14(5)^{2}

A = 3.14(25)

A = 78.5 cm^{2}

The circumference of a circle = 2πr = 2(3.14)(5)

Circumference = 3.14(10) = 31.4 cm.

Hence, the perimeter and area of circle are 31.4 cm and 78.5 cm^{2} respectively.

**Example 1:** If the radius of a circle is 28 cm find the circumference of the circle.

**Solution:**Given, Radius of the circle = 28 cm. To find the circumference of the circle, we will use the circumference formula: 2πr = 2 × 22/7 × 28 = 176 cm.Therefore, the circumference of the circle is 176 cms.

**Example 2:** The circumference of a wheel is 440 cm. Find its radius and diameter.

**Solution:**Given, the Circumference of the wheel = 440 cmCircumference of a circle formula = 2πrLet us substitute the known values to find the radius first.440 = 2πr440 = 2 × (22/7) × rradius = 70 cmDiameter = 2 × radiusDiameter = 2 × 70Therefore, the radius is 70 cm, and the diameter is 140 cm.

**Example 3: **The perimeter of a rectangular wire is 264 m. The same wire is bent into the shape of a circle. Find the radius of the circle formed using the circumference formula.

**Solution:**We know that the perimeter of the rectangle = Total length of the wire used = Circumference of the circle formed.Hence, the Circumference of the circle formed = 264 mCircumference of a circle formula = 2πrCircumference of the circle = 264Let us substitute the known values to find the radius.264 = 2πr264 = 2 × (22/7) × rTherefore, the radius of the circle is 42 m.

### Practice Questions

- Calculate the perimeter of circle whose diameter is 8 cm.
- What will be the diameter of a circle if it’s C = 10 cm?
- If C = 12 cm, what will be its radius?
- What is the circumference of a 16-inch circle?
- What is the circumference of a 6 mm circle?

**Frequently Asked Questions on Circumference of a Circle**

Q1

### What is the Circumference of a Circle?

The circumference of a circle is defined as the linear distance around it. In other words, if a circle is opened to form a straight line, then the length of that line will be the circle’s circumference.

Q2

### How to Calculate the Circumference of a Circle?

To calculate the circumference of a circle, multiply the diameter of the circle with π (pi). The circumference can also be calculated by multiplying 2×radius with pi (π=3.14).

Q3

### How to Calculate Diameter from Circumference?

The formula for circumference = diameter × π

Or, diameter = circumference/π

So, the diameter of the circle in terms of circumference will be equal to the ratio of the circumference of the circle and pi.

Q4

### What is the Circumference of a Circle with Radius 24 inches?

Circumference = 2×π×r

C = 2×3.14×24

C = 150.72 inches

Q5

### What are the steps involved in finding the circumference of a circle if its area is given?

Step 1: Find the radius using the area of the circle formula (A = πr^{2}).

Step 2: Now, substitute the radius value in the circumference formula (C = 2πr) to get the answer.

Q6

### Which formula is used to find the perimeter of a circle, when its area is given?

If the area of a circle is given, then the formula to calculate the perimeter or a circumference of a circle is given by: C = 2√(πA) units.

Q7

### Find the circumference of a circle in terms of π, if the radius measures 3 cm.

We know that, Circumference = 2πr = 2π(3) = 6π

Therefore, the circumference of a circle is 6π cm, if its radius is 3 cm.

Q8

### What is the circumference of a circle in terms of π, if its diameter is 7 cm.

As we know, Circumference of a circle if its diameter is given = π × d units.

Substituting d = 7 cm in the formula, we get

C = π × 7 = 7π units.

## What is Circumference of a Circle?

The **circumference** of a circle is its boundary or the length of the complete arc of a circle. Let us understand this concept using an example. Consider a circular park shown below.

If a boy starts running from point ‘A’ and reaches the same point after taking one complete round of the park, a distance is covered by him. This **distance** or **boundary** is called the **circumference** of the park which is in the shape of a circle. The circumference is the length of the boundary.

### What is the Circumference of a Circle in Geometry?

The **circumference of a circle** is the measure of the boundary or the length of the complete arc of a circle. The circumference of the circle is the product of π (pi) and the diameter of the circle. The circumference of a circle is a linear quantity that has the same units of length.

### How to Find the Circumference of a Circle?

The circumference of a circle is calculated with the help of the circumference formula that needs the value of the radius of the circle and the value of π (pi). Circumference of a circle = 2πr, where, ‘r’ is the radius of the circle and π(pi) is a special mathematical constant with a value approximated to 3.14159 or π = 22/7.

### How to Find the Diameter From the Circumference of a Circle?

If we need to calculate the diameter when the circumference of a circle is given, we use the formula: Circumference = π × Diameter, or, Diameter = Circumference/π

### How to Find the Circumference of the Circle with the Area?

The circumference of a circle can be calculated if the area of the circle is given. Using the formula for the area of a circle the radius can be calculated. Once the radius is known, the circumference can be calculated.

### What is the Unit of the Circumference of a Circle?

The circumference of the circle is a one-dimensional linear quantity and the unit of the circumference of a circle is expressed in m, inch, cm, feet, and so on. The circumference of a circle is related to other linear quantities such as the radius and diameter of the circle.

### What is the Perimeter of the Circle?

The perimeter of a circle is the same as the circumference of a circle. It is the total length of the outer boundary of the circle. The perimeter or circumference of a circle is the product of the constant ‘pi’ and the diameter of the circle. It is expressed in linear units like m, inch, cms, feet.

### What is the Value of Pi?

Pi is a constant value used for the measurement of the area and circumference of a circle or other circular figures. The symbol of pi is π and its numeric value is equal to 22/7 or 3.14. Further, these numeric values are used based on the context of the equation.

### What is the Difference Between the Diameter and the Circumference of a Circle?

The diameter of the circle is the longest chord that passes through the center of the circle. The circumference of the circle is the length of the outer boundary of the circle. Both the diameter and the circumference are lengths and are expressed in linear units. The circumference of the circle is equal to the product of the diameter and the constant π (pi).

### How to Find the Circumference of a Circle with Diameter?

The circumference of the circle can be calculated if the diameter is known because the relationship between the circumference and diameter of the circle is expressed as, Circumference = π × Diameter, or, diameter = Circumference/π.