Circle Formula: Definition, Properties with Solved Examples

Get Circle Formula with examples from our best teachers. You can learn everything related to circle formulas like Radius, Area, and Circumference. In our day – to day life, we can come across many objects which are round in shape, such as dials of many clocks, wheels of a vehicle, bangles, key rings, coins of denomination rs 1, 2,5,10, 20, etc. 

In a clock, we observe that the second’s hand goes around the clock’s dial rapidly, and its tip moves in a round path. This path traced by the direction of the second hand is called a Circle. 

Circle 

A circle is a collection or cumulation of all the points. For example, suppose P is a point in a plane, each at a constant or fixed distance from a point in that plane.

In other words, a Circle is the path of a particular point that moves in a plane so that it remains at a const. distance from a fixed point.

Here,

• ‘ r ‘ denotes the radius of the circle.
• ‘ d ‘ indicates the diameter of the circle.
• ‘ c ‘ indicates the circumference of the circle.

Parts of Circle 

A circle has many different parts depending on its position and properties. Other features of a circle are explained in detail below.

• Centre- It is the midpoint of a circle. 
• Sector – A sector is defined as the region or area bounded by the two radii and the intercepted arc of the circle.
• Radius– Radius is the distance from the center point to the circle’s edge.
• Annulus-The region bounded by two concentric circles. It is a ring-shaped object. See the figure below.
• Arc – An arc is defined as a part of the boundary of a circle or curve. It may also be referred to as an open curve.
• Segment- A circle segment can be defined as the area bounded by a chord and the corresponding arc lying between the chord’s endpoints.
• Secant- A straight line that cuts a circle at two points is called a secant.
• Tangent- A tangent is a line or ellipse that touches itself at only one point.
• Chord- A line segment joining two ends of a circle is called a chord.

Formulas Related to Circles

We know that a circle is a 2-dimensional curve-shaped figure, and the two different parameters are used to measure the ring.

The formulas of the Circles are given below:

Diameter of a Circle

D = 2× r

The radius of a Circle

We know that,

D= 2× r

hence, R= D/2

Area of a Circle

A= π × r2

Circumference of a Circle

C= 2 × π × r

Properties of Circles

There are many properties related to a circle; some of the basic properties of the circle are given below:

• The circle’s diameter is the major chord and doubles the radius.
• Circles of equal radius are congruent to each other.
• Circles that differ in size or have different radii are similar.
• The diameter of a circle divides it into two equal parts.
• In a circle, the outermost line is equidistant from the center of the process.

Solved Examples related to Circle:

Example- 1

A circle has a radius of 10 cm. Calculate its diameter, area, and circumference.

Solution:

Given parameters are,

Radius, r = 10 cm

The diameter of a circle is- 

2r = 2 × 10 = 20 cm

r= 10 cm

The area of a circle is- 

π r2 

= π × 100

= 314 cm2

Circumference of a circle is given by

2 π r

= 2 × π × 10

= 62.8 cm ( Answer )

Example- 2

Find the area and the circumference of a circle whose radius is 4 cm. (Take the value of π = 3.14)

Solution:

Given: Radius = 4 cm.

Area =π r2 

= 3.14 × 42

A= 50.24 cm2

Circumference, C = 2πr

C= 2 ×3.14× 4

Circumference= 25.12 cm ( Answer )

Example- 3

Find the circular park area with a radius of 100m.

Solution:

To find the area of a park.

Given:

The radius of the park= 100 m

Using all circle formulas (the area of a circle formula is),

The area of a Circle is= π × r2

= π × 1002

= π × 10000

=31400 m2 ( Answer )

Example 4:

The radius of a circle is 6 in. Using circle formula, calculate the circumference of the circle.

Solution:

To find: the circumference of a circle

Given: r = 6in

The perimeter of circle formula = 2 π r

C = 2 × (22/7) × (6)

C = 37.68 inches. ( Answer )

Extra Examples Related to Circle

Example 5:

Using the perimeter formula of a circle, find the circle’s radius having a circumference of 200 inches.

Solution: 

To find: Radius of the circle

Given: Circumference = 200 in

Using the perimeter formula of a circle,

The perimeter of the circle or the circumference = 2 π r

2 π r = 200

2 × 22/7 × r = 200

r = 200 × 7 / 44

r = 31.81 inches ( Answer )

Example 6: 

Find the circle’s area whose circumference is 62.8 cm.

Solution:

Given:

Circumference = 62.8 centimeters 

To find the circle’s area, we need to find the radius.

From the circumference, the radius can calculate:

2 π r= 62.8

(2)(3.14)r= 62.8

r= 31.4 /(2)(62.8)

r=20/2

r= 10 cm

Therefore, the radius of the circle is 10 cm.

The area of a circle is πr2 square units.

Now, put the value of the radius in the area of a circle formula, and we get

A= π(10)2

A= 3.14 x 100

Therefore, Area= 314 cm2

Therefore, the area of a circle is 314 cm2. ( Answer )

Example 7:

Find the area, circumference, and diameter of a circle of radius 20 cm.

Solution:

Given,

The radius of a circle, r= 20 cm

Diameter of a circle given by

2r

= 2 × 20

= 40 cm

The area of a circle is- 

π r2

= π ×402

= π × 1600

Therefore, π r2= 5024 cm2

Circumference of a circle

= 2 π r

= 2 × π × 20

Circumference of a circle= 125.6 cm ( Answer )

Practice Problems on Circle:

You can Solve the following circle problems by yourself, which are given below:

1. Find the circumference of a circle whose radius is 28 m.
2. Find the area of a circle whose radius is 13 cm.
3. The area of a circle is 286 cm2. Find its radius.
4. Find the circular park area with a radius of 275 m.
5. Using the perimeter formula of a circle, find the circle’s radius with a circumference of 150.

Frequently Asked Questions on Circles:

1. Define a circle?

Answer: A circle is a closed 2-dimensional curve-shaped figure, where all the points on the surface of the circle are equidistant from the middle point.

2. What is the Diameter of the Circle? 

Answer: A chord of a circle that passes through its center, called a Diameter of the Circle.

Hence, Diameter, D= 2r,

Here, r is the radius of a circle.

3. What is the sector of the Circle? 

Answer: The part of the plane region enclosed by an arc of a circle and its two bounding radii is called the sector of the Circle.

4. What is the Circumference of the Circle? 

Answer: The whole arc of a circle is called the Circumference of a circle. 

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