Circle Definition

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Definition of the Circle

In mathematics, Circle is a two-dimensional figure which is round in shape. All the points on the boundary of the circle are equidistant from a point inside the circle which is known as the center of the circle.

 

The distance between the center of the circle to the point on its boundary is termed as the radius of the circle which is denoted as “r”.


The double of the radius of the circle is the diameter of the circle which is the largest chord of any of the circles.


Diameter = 2×Radius

The semicircle is half of the circle and the quadrant of the circle is the one-fourth part of the circle.


Parameters and the Formulas of the Circle:

Radius

The distance between the center of the circle to the boundary of the circle is called the radius of the circle.


Chord

A straight line of any length that cuts the boundary of the circle from both ends is called the chord of the circle. The largest chord of the circle is the diameter of the circle.


Secant

Secant is the line segment that cuts the circle into two segments which are termed as the minor segment and the major segment of the circle.


Tangent

Tangent is the line segment that touches the circle on its circumference only and that is always perpendicular to the radius of the circle.

You can refer to the following image to locate the mentioned parameters of the circle.

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Circumference of the Circle

The circumference is the length of the boundary of the circle which can be calculated as:

Circumference = 2πr

Where the value of π can be \[\frac{22}{7}\] or 3.14


Area of the Circle:

The area of the circle is the inner part of the circle which can be calculated by the mentioned formula.

Area = πr2


Example: 

If the diameter of the circle is 21 cm then find the circumference of the circle.


Solution: 

we have given the diameter of the circle as 21 cm,

Then the radius of the circle will be

Radius = 21/2

Circumference of the circle is given as:

Circumference = 2πr 

Circumference = 2 x \[\frac{22}{7}\] x \[\frac{21}{2}\]

Circumference = 22×3

Circumference = 66 cm


Example: What is the area of the circular park, if the radius of the park is 7m?

Solution: The radius of the circular park is 7 m

The area of the park is given by:

Area = πr2

Area = \[\frac{22}{7}\] x 7 x 7

Area = 22×7

Area = 154 cm2

So the area of the park will be 154cm2.


Incircle Meaning

When the circle is inscribed in any of the polygons, it is called the incircle of the polygon, where the sides of the polygon are the tangent of the circle. Let us understand the exact definition of the incircle by taking the polygon as a triangle.


Incircle Definition

Incircle in the triangle is formed by taking the intersection point of the angle bisector of the angles of the triangle as the center. From that point, we get the circle of the radius termed as the inradius, which is called the incenter of the circle.

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Where the inradius is given by:

r = \[\frac{\Delta }{s}\]

Where Δ is the area of the triangle and s is the semi perimeter of the triangle.


Special Cases of the Inradius of the Incircle


Case 1

If the triangle is the equilateral triangle, then the inradius of the circle is given by:

The side of the equilateral triangle will be a

r = \[\frac{\Delta }{s}\]

As we know that the area of the equilateral triangle is \[\sqrt{\frac{3a^{2}}{4}}\] 

And the semi perimeter will be \[\frac{3a}{2}\] 

So the inradius will be

r = \[\frac{\Delta }{s}\]

r = \[\frac{\sqrt{\frac{3a^{2}}{4}}}{\frac{3a}{2}}\]

r = \[\frac{a}{2\sqrt{3}}\]


Case 2

If the triangle is right-angled, then the inradius of the incircle is given by:

r = P+B-H/2

Where P, B, and H are the perpendicular, base, and hypotenuse of the right-angled triangle respectively.

FAQ (Frequently Asked Questions)

Q1. What is the Definition of the Circumference of the Circle? Find the Circumference of the Circle if the Radius of the Circle is Given as 14 cm.

The circumference of the circle is defined as the length of the boundary of the circle. It can also be defined as the length of the arc of the circle which is also known as the perimeter of the circle.

The circumference of the circle is calculated as:

Circumference = 2πr

Where the value of π can be 22/7 or 3.14

The solution to the problem:

we have given the radius of the circle as 14 cm,

Circumference of the circle is given as:

Circumference = 2πr 

Circumference = 2 × 22/7 × 14 

Circumference = 2×22×2

Circumference = 88 cm

Hence the circumference of the given circle will be 88 cm.

Q2. What Do You Mean By the Incentre of the Circle? What is the Inradius of the Circle if You Have Given the Equilateral Triangle of the Side 8 cm?

Incircle in the triangle is formed by taking the intersection point of the angle bisector of the angles of the triangle as the center. From that point, we get the circle of the radius termed as the inradius which is called the incenter of the circle. The inradius of the circle is given by:

r = Δ/s

If the triangle is the equilateral triangle, then the inradius of the circle is given by:

The side of the equilateral triangle will be a

Then inradius will be

r = a/2√3

We have given the side of the triangle as 8 cm then the inradius will be

r = a/2√3

r = 8/2√3

r = 4/√3

Hence the inradius of the given circle will be 4/√3 cm.