A circle is a labeled by its center point: O means the circle with center point O
A line segment, generally denoted by the variable, that connects two points on the circles and passes through the center of the circle
Radius:- A line segment, generally denoted by the variable r from the center of the circle to any point on the circle.
Chord: A line Segment joining two points on the circle segment C is a chord.
Tangent:- A line that touches only one point on the circumference of the circle. A line drawn a tangent to a circle is perpendicular to the radius at the point of contact.
Circumference:- The distance around a circle is called the circumference.
An Arc is a portion of the circumference of a circle. The shorter distance between A an B along the circle is called the minor arc.
The longer distance A and B is the major arc.
Arc length:- For an arc with a central angle measuring θ Degree:
Length = ( θ / 360 ) (Circumference)
= (θ / 360 ) ( π d )
Example: What is the length of arc Abc of the circle with center O shown?
Ans:- Arc length= (Q / 360 ) ( π d )= (60 / 360 ) (π (2))
12 π / 6 = 2 π
Area of circles: The area of circles is given by the formula A= π r2
A sector is a portion of the circle that is bounded by two radii and an arc.
In a sector Whose central angle measures by Q degrees.
Area of sector= (Q / 360) x (Area of circle)
Example:- What is the area of sector AOC in the circle with the center O shown?
Ans) Area of sector AOC= ( 60 / 360 ) (36π ) = ( 6 π )
- The perpendicular from the center of a circle to a chord bisects the chord and vice versa.
If ∠OCB=90 °, THEN AC=BC
If AC=BC , then ∠OCB= 90°
2) Equal chords of a circle are equidistant from the center. Conversely, chords equidistant from the center are always equal.
3) Any two angles in the same segment are equal. Thus ∠ACB = ∠ADB
4)The angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc at any other point on the circle.
5)The angle subtended by a semicircle is a right angle. Conversely, the arc of a circle subtending a right angle at any point on the circle is a semi-circle.
If AB is a diameter, then ∠ACB = 90°
IF ∠ACB = 90° , then AB is a diameter.
6)Tangent drawn from common external point to a circle are equal.
Q):- A 5 by 12 rectangle is inscribed in a circle. What is the circumference of the circle?
BD is a diameter (Because The arc of a circle subtending a right angle at any point on the circle is a semi-circle)
In BCD , BC2 + DC2 = BD2 (by Pythagoras theorem)
BD = 13
Circumference of the circle = 2π (Radius) = 13 π
Q) In the figure shown below, if the radius of a circle with centre P is three times the radius of circle with centre A, ∠BAC=∠QPR, and the shaded area of the circle with centre A is 3 π SQUARE units, then what is the area of the shaded part of circle with centre P?
Ans) Let ∠BAC =∠QPR = Q°
The shaded area of circle A = 3 π sq. units = ( Q / 360° ) π (AC)2
PQ = 3AC
= ( Q / 360° ) π (PQ)2
= ( Q / 360° ) π (3AC)2
= (3π x 9)
= 27π sq. units
Ques 28: -Two congruent, adjacent circles are cut out of a 16 by 8 rectangle. The circles have the maximum diameter possible. What is the area of the paper remaining after the circles have been cut out?
Ans) For the circles, the diameter of the circle is the same as the width of the rectangle.
Remaining area = Area of rectangle – 2 x area of circle
Area of rectangle = Length x Breadth = 8 x 16 = 128
radius of circle = 4
Area of circle = π (4)2 = 16 π
Remaining area = (128- 2x(16π) ) = ( 128 – 32 π) sq. units
Ques: The Figure shows an equilateral triangle, where each vertex is the center of a circle. Each circle has a radius of 20. What is the area of the shaded region?
Ans:- Side of equilateral triangle =40
Area of equilateral triangle= √3/4 (40)2
=√3/4 (1600) = (400 √3)
Area of three 60° sectors= 3 x (60/360) (π) (r2)
=(1/2) π (400) = 200π
Area of shaded Region= Area of equilateral triangle- area of three 60° degree sectors
= (400√3 – 200 π ) sq. units