Properties of Circle

In geometry, lines and circles are the most basic figures. A line is a locus of a point moving in a constant direction, but a circle is a locus of a point moving at a constant distance from a fixed point, as we all know. The circle's theoretical significance is reflected in the quantity of spectacular applications. In this article, we will discuss all properties of circle, semicircle, area, the circumference of a circle and a few solved examples.

Polar Moment of Inertia Circle

The resistance given by a beam or shaft when twisted by torsion is known as the polar moment of inertia. This opposition is usually caused by the cross-sectional area, and it is important to highlight that it is unaffected by material composition. The torsional resistance of the object will be stronger if the polar moment of inertia is larger. To turn the shaft at an angle, more torque will be necessary.

The following formula is commonly used to calculate a circle's moment of inertia or second-moment area:

I=R44

The radius is R, and the axis passes through the centre. This equation is equivalent to I=D464

when we express it in terms of the diameter (D) of the circle.

Similarly, the moment of inertia of a circle about an axis perpendicular to its circumference is equal to;

I=5R44

The polar moment of inertia circle is expressed as;

I=5R42

Semi Circle

For a semi-circle the formula is given as;

I=R48

Angle Properties of a Circle

Following are the Angle Properties of a Circle:

1. The inscribed angle of an arc at its centre is twice the angle formed by the same arc.

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Central Angle = Twice the inscribed angle

∠AOB=2∠ACB

ii. Angles subtended by diameter at the circumference are always 90°.

iii. Angles subtended by the same arc at the circumference of a circle are equal.

iv. In a cyclic quadrilateral, the angles add up to 360°.

Properties of Semicircle

• A semicircle is formed when a line passing through the centre touches the two ends of the circle.

• When a circle is cut into two halves or when the circumference of a circle is divided by 2, we get a semicircular shape.

• Since a semicircle is half that of a circle, hence the area will be half that of a circle.

• The area of a circle is the number of square units contained within it.

• A semicircle's surface area is half that of a whole circle.

• The angle inscribed in a semicircle is always 90°.

• The inscribed angle is formed by drawing a line from each end of the diameter to any point on the semicircle. Regardless of the arc's length, the angle formed where two lines meet the arc will always be 90°.

• The inscribed angle and the two extremities of the semicircle's diameter will always create a right triangle inside the semicircle.

Circle Definition

A circle is a collection of points in which all of the points are equidistant from the centre. The following are some of the key terminologies used while referring to circle:

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Circumference - The circumference is the circumference of a circle.

Radius- The radius is the line that runs from the circle's centre "O" to its circumference, and it is symbolised by the letter "r."

Diameter - A diameter is a chord that travels through the centre. Every diameter has a length equal to twice the radius because it is made up of two radii linked at their extremities. The term "diameter" is applied to both these intervals and their common length. The letter "D" stands for it.

Arc -The arc of a circle is defined as the part or segment of the circumference of a circle. The major arc is the greatest arc on the circumference, while the minor arc is the smallest.

Sector - The region contained by two radii and the associated arc in a circle is known as the sector of a circle. Sectors are divided into two categories: minor and major.

Chord - A chord is a straight line that connects any two points in a circle.

Tangent - It's a line that only touches the circle's circumference at one place

Secant - A line that cuts a circle at two distinct points is called a secant. A chord is an interval cut off by a secant going through the centre of a circle, while the diameter is an interval cut off by a secant passing through the centre of a circle.

Properties of Circles Geometry

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From the Above Diagram, We Will Write the Geometrical Property of the Circle

• Radius is of the same length. So OA=OC. 

• The angle at the centre is twice the angle formed at the circumference:2∠ABC= ∠AOC

• Inscribed angle theorem:∠ABC=∠APC

• Alternate segment theorem: ∠ABC=∠ACT

• Tangents to the circle from a point will have the same length: TA =TC.

• Angles subtend equal arcs and vice versa.

• Equal chords rest on equal angles, and vice versa.

• Equal chords and vice versa are equidistant from the centre.

• A chord's perpendicular bisector crosses through the circle's centre.

• A unique circle is formed by any three non-colinear points.

Circle Properties

Some of the Important Properties of the Circle are Given Below:

• The circumference of equal chords and equal circles is the same.

• The chord is bisected by a radius drawn perpendicular to it.

• Circles having different radii are similar.

• A rectangle, trapezium, triangle, square, or kite can all be circumscribed by a circle.

• The distance between the circle's centre and the longest chord (diameter) is zero.

• When the chord length is increased, the perpendicular distance from the circle's centre decreases.

• If we draw the tangents at the end of the diameter, then they are parallel to each other.

• The radii connecting the ends of a chord to the centre of a circle form an isosceles triangle.

Circle and Its Properties

Following are Some of the Most Important Properties of Circle:

• Congruent circles are those with equal radii.

• The diameter of a circle is the circle's longest chord.

• The perpendicular drawn from the centre of the circle on any given chord of a circle divides the chord into two equal halves.

• All chords of a circle that are equidistant from the circle's centre are the same length.

• If we draw two tangents to a circle from a given location outside the circle, they will be of equal length.

• A right angle is formed by a tangent to a circle and the radius of the circle at the point of contact of the tangent.

• 360° is the angle formed in the centre of a circle by the circumference of the circle.

• A circle's radius is proportionate to its circumference.

• Inside a circle, we can draw a triangle, square, trapezium, rectangle, or kite with all of their vertices on the circumference of the circle.

• The length of a circle's chord is inversely proportional to its perpendicular distance from the circle's centre.

• We can encircle a triangle, square, or kite with a circle.

• If we draw two tangents drawn to a circle at the two endpoints of a diameter are parallel to each other.

Circle Tangent Properties

Following are the Properties of Tangent to a Circle:

• The tangent always makes single point contact with the circle.

• At the point of tangency, it is perpendicular to the circle's radius.

• It never intersects the circle at two points.

• Tangents from an exterior point to a circle have the same length.

Concyclic Points Properties

• If a circle is drawn through all of the points in a set, it is said to be concyclic.

• The four points are said to be concyclic if the segment connecting two points A and B subtends equal angles at two other points C and D on the same side of AB.

• Three non-collinear points are always concyclic, and they are all connected by a single circle.

Properties of Incircle

• The centre of the circle is called the incenter of the triangle when it is inscribed in a triangle so that it touches each side of the triangle.

• If I is the triangle's incenter, the line segments AE and AG, CG and CF, BF and BE are all the same length.

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•  If I is the incenter of the triangle, then ∠BAI=∠CAI, ∠ABI=∠CBI and ∠BCI=∠ACI

• The sides of the triangle are tangents to the circle, hence OE=OF=OG=r are called the inradii of the circle.

• The coordinates of incenter of the triangle ABC with coordinates of the vertices,A(x1,y1),B(x2,y2)and C(x3,y3)and a,b,c are sides

(ax1+bx2+cx3a+b+c,ay1+by2+cy3a+b+c)

Circle Formulas

Area of a circle, A = r2 square units

The circumference of a circle = 2πr units

The circumference of a circle formula is  πd

Where,

Diameter = 2 x Radius

D = 2r

Here “r” represents the radius of a circle.

Solved Examples

1. Find the Area and Circumference of a Circle Having a Diameter Value of 10 cm?

Ans: Given:

Diameter, D = 10 cm

We know that diameter = 2 x Radius

Therefore, radius, r = d/2

r = 10/2 = 5

So, the radius is 5 cm.

Area of a circle, A = r2 square units

A = 3.14 x 5 x 5

Where ,

π = 3.14

A = 3.14 x 25

A = 78.5 cm2

Hence, the area of a circle is 78.5 cm2

The circumference of a circle = 2πr units

C = 2 x 3.14 x 5

C = 10 x 3.14

C = 31. 4 cm

Therefore, the circumference of a circle = 31.4 units.

2. Calculate the Circumference of a Semicircle That has a Radius of 7 cm.

Sol: Circumference of a Semi-circle = πr + 2r

= [(22/7) x 7] + (2 x 7)

= 22 + 14

= 36 cm.

Hence the circumference of a semicircle is 7 cm.