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.
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:
Area of a circle (A ) = π r 2 =( π/4 ) D2 = 0.7854 D2
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
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 ]
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