## Properties of circle in math | Arc, Perimeter, Segment of circle

A circle can be defined as, it is the locus of all points equidistant from a central point. In this we discuss about *Properties of circle*, *circle formulas like area*, **perimeter, arc length, segment length, segment area..**. etc.

### Terminology related to circles in math:

**Origin : **It is a center(equidistant) point of the circle. Here “O” is the *origin of the circle.*

**Radius :** Distance from center of circle to any point around it **circumference** is called *Radius of the circle.* Generally it is denoted by ” r “.

**Diameter: **The longest distance from one end of a circle to the other end of the circle is called dia of the circle. Generally it is denoted by ” D ” . Diameter of the circle = 2 x Radius of the circle. i. e D = 2r.

**Arc of a circle:** It is a part of the circumference of the circle. The bigger one is called the major arc and the smaller one the minor arc.

**Sector of a circle:** It is a part of the area of a circle between two radii (a circle wedge).

**Chord :** A line **segment** within a circle that touches two points on the circle is called *chord of a circle.*

**Circumference :** The distance around the circle is called* circumference or perimeter of the circle.*

**Pi ( π ****):** It is a number equal to **3.141592… or 22/7.**

The **pi ( π ****)** = (The circumference) / (The diameter) of any circle.

**Tangent of circle:** a line perpendicular to the radius that touches ONLY one point on the circle.

**Secant of circle** : A line that intersects a circle at two points then it is called *Secant of circle*.

### Properties of circle :

**Congruency**: Two circles can be congruent if and only if they have**equal radii.**- The perpendicular from the center of a circle to a chord
**bisects the chord**. The converse is also true. - The perpendicular bisectors of two chords of a circle intersect at its center.
- There can be one and only one circle passing through three or more non -collinear points.
- If two circles intersect in two points then the line through the centers is the perpendicular bisectors of the common chord.
- If two chords of a circle are equal, then the center of the circle lies on the angle bisector of the two chords.
- Equal chords of a circle or congruent circles are equidistant from the center.
- Equidistant chords from the center of a circle are equal to each other in terms of their length.
- The degree measure of an arc of a circle is twice the angle subtended by it at any point on the alternate segment of the circle.
- Equal chords of a circle ( or of congruent circles) subtended equal angles at the center.(at the corresponding centers) The converse is also true.
- If the sum of the opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic.
- Secant means a line that intersects a circle at two points. Tangent means it is a line that touches a circle at exactly one point.
- In two concentric circles , the chord of the larger circle that is tangent to the smaller circle is bisected at the point of contact.

### Circle Formulas in Math :

#### Area and circumference of a circle:

Here Origin of the circle = O , Diameter = D and Radius = r

**Area of a circle (A )** =** π r ^{2} =( π/4 ) D^{2} = 0.7854 D^{2}**

**Circumference of a circle ( C )**=

**2 π r = π D.**

*Area of circle =( 1/2) x Circumference x radius*

**A = (1/2) x C x r**

**Diameter of a circle (D)** =** √(A/0.7854).**

#### Arc and sector of a circle:

Here angle between two radii is ” θ” in degrees. . And sector of a circle AOB.

**Arc length of circle**( l ) (minor) =** ( θ /360) x 2 π r = θ π r / 180**

**Area of the sector **(minor) =** ( θ /360) x π r **^{2 }

If the angle θ is in radians, then

*The area of the sector* = **(θ/2) r ^{2}**

*Sector angle of a circle θ* = **(180 x l )/ (π r ).**

#### Segment of circle and perimeter of segment:

Here radius of circle = r , angle between two radii is ” θ” in degrees.

*Area of the segment of circle = Area of the sector – Area of ΔOAB.*

**Area of the segment** =** ( θ /360) x π r ^{2 } – ( 1 /2) x sinθ x r ^{2 }**

**Perimeter of the segment** =** (θ π r / 180) + 2r sin (θ/2).**

**Chord length of the circle** = **2 √ [ h (2r – h ) ] = 2r sin (θ/2).
**

**Arc Length of the circle segment ** = l = **0.01745 x r x θ**

Online calculator for circle segment area calculation

#### Area of the circular ring:

Here big circle radius = R and Dia = D,

Small circle radius = r and Dia = d,

**Area of a circular ring ** =** 0.7854 (D ^{2} – d ^{2}) = (π/4) ( D ^{2} – d ^{2})**

*Area of a circular ring* =** π (R ^{2} – r ^{2 }).**

#### Formula for intersecting chords in circle:

Here AB and CD are two chords in circle and intersecting each at the point E.

Then AE : EB = DE : EC.

#### Formula for length of the tangents of circles:

Here Two circles origins O & O’ and radius are r1 and r2 respectively.

*Direct common tangent AB & transverse common tangent = CD*

**Length of direct common tangent AB =** **√ [ (Distance between two origins) ^{2} – (r1 -r2)^{2} ]**

= **√ [ (OO’) ^{2} – (r1 -r2)^{2} ]**

**Length of transverse common tangent AB** = **√ [ (Distance between two origins) ^{2} – (r1 +r2)^{2} ]**

= **√ [ (OO’) ^{2} – (r1 +r2)^{2} ]**

### Geometry Math

Two dimensional shapes formulas.

Quadrilateral Properties | Trapezium, parallelogram, Rhombus

Types of Triangles With examples | Properties of Triangle

**Number System.**

Rules for Divisibility of numbers

Formulas for Sum of n Consecutive numbers

GCD and LCM Problems & Solutions

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## 12 thoughts on “Circle formulas in math | Area, Circumference, Sector, Chord, Arc of Circle”

## Steve LeVine

(December 31, 2018 - 10:05 pm)There is an error:

“Area of the segment = ( θ /360) x π r 2 + ( 1 /2) x sinθ x r 2”

…should be “Area of the segment = ( θ /360) x π r 2 – ( 1 /2) x sinθ x r 2”

## sivaalluri

(January 22, 2019 - 3:31 pm)Thank you for your valuable correction

## Lydia

(August 1, 2019 - 9:45 pm)Your post was quite helpful…… would’ve been better if you added actual examples with figures.

## sivaalluri

(August 2, 2019 - 2:23 pm)Ok thank you for your suggestion

## Vinit

(December 24, 2019 - 2:47 am)i think I invented a math formula

What should I do now

## sivaalluri

(December 29, 2019 - 1:59 pm)Write in comment of your invented formula

## Augusto

(April 21, 2020 - 3:32 am)How can I find the angle Theta or height h knowing the area of the segment A?

## sivaalluri

(April 26, 2020 - 11:48 am)Area of the segment=( θ /360) x π r^{2 }– ( 1 /2) x sinθ x r^{2 }## SCOTT

(August 5, 2020 - 11:38 pm)I have a problem, I cannot remember the formula for finding the length of an arc. I have a chord that is AB=20ft. from the center of the chord to the center of the arc CD=10ft. I need the formula to find the length of the arc EF=?. I need the numerical value of the letters put into the equation so that I understand it. You know higher math for dummies been out school for 55 years.

Thanks

Scott

## Ashok Kumar Bandyopadhyay

(July 19, 2021 - 3:59 am)Determine the areas just outside the two chords separated by a distance of 0.95d of a circle of diameter d.Send the answer to my mail address with the method of calculation. I have no website.

## Peter hakim

(August 31, 2022 - 9:10 pm)Thanj you for your help

## sivaalluri

(October 14, 2022 - 5:38 am)welcome