Chapter 12 Areas Related to Circles
12 Areas Related to Circles
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Fundamentals: Circle is defined set of all those points which are at constant distance from fixed point. The fixed point is called centre. The constant distance is called radius. The longest chord passing through centre and whose end point lies on circle is called diameter. Circles with same centers are called Concentric Circles. Circumference: In simple words we can say perimeter of circle is called circumference. But actually p is defined as the ratio of circumference and diameter of circle.
Circumference Diameter Therefore, Circumference = π × Diameter If r is the radius of a circle, then (i) Circumference = 2πr or πd, where d = 2r is the diameter of the circle. 2 2 (ii) Area = πr or πd /4 p=
(iii) Area of semi-circle =
pr 2 2
(iv) Area of a quadrant of a circle =
pr 2 4
Area enclosed by two concentric circles: If R and r are radii of two con-centric circles, then area enclosed by the two circles 2 2 2 2 = πR – πr = π (R – r ) = π (R + r) (R – r)
r
R
Some useful results: (i) If two circles touch internally, then the distance between their centres is equal to the difference of their radii. (ii) If two circles touch externally, then the distance between their centres is equal to the sum of their radii. (iii) Distance moved by a rotating wheel in one revolution is equal to the circumference of the wheel. (iv) The number of revolutions completed by a rotating wheel in one minute
=
Distance movedin oneminute Circumference
Arc, Chord, Segment, Sector of a Circle
Q M ajor S ector
M ajor S egm ent
O r A
M inor S egm ent
B
q
r
M inor S ecto r
A
B
P Arc: Any portion of circumference. e.g.: APB is minor arc while AQB is major arc. Chord: The line joining any two points on the circle. e.g.: AB. Segment: In figure chord AB divides the circle in two segments i.e., APBA (minor segment) and AQBA (major segment). Sector: The region bounded by the two radii AO and BO and arc AB is called sector of the circle. Let ∠AOB = ?, where ? is called central angle. Length of Arc: When sector angle ∠AOB = θ. We know that length of arc when sector angle (∠AOB = 360°) is 2πr length of arc when sector angle (∠AOB = 1°) is =
2 pr 360°
Length of arc when sector angle (∠AOB = θ) is =
2 pr ´ q 360°
Length of arc AB = 2pr ´ Perimeter of the sector = 2r ´
q 360°
q ´ 2pr 360°
Area of Sector: When sector angle ∠AOB = θ 2 We know that area of circle when sector angle (∠AOB = 360°) is πr pr 2 360°
Area of arc when sector angle (∠AOB = 1°) is = Area of arc when sector angle (∠AOB = θ) is =
pr 2 ´q 360°
q 360° Area of Segment (shaded) of A Circle : Area of Sector AOB – Area of triangle AOB 2 Area of Sector = pr ´
We can use formula given below when q ³ 90° pr 2 q q q Area of segment = - r 2 sin cos 360° 2 2
We can use formula given below when θ = 90° pr 2 q 1 2 Area of segment = - r sin q 360° 2
Perimeter of Segment (shaded) of A Circle: AB + arc (APB) Perimeter of Segment =
q 2prq - 2r sin 360° 2
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Fundamentals: Circle is defined set of all those points which are at constant distance from fixed point. The fixed point is called centre. The constant distance is called radius. The longest chord passing through centre and whose end point lies on circle is called diameter. Circles with same centers are called Concentric Circles. Circumference: In simple words we can say perimeter of circle is called circumference. But actually p is defined as the ratio of circumference and diameter of circle.
Circumference Diameter Therefore, Circumference = π × Diameter If r is the radius of a circle, then (i) Circumference = 2πr or πd, where d = 2r is the diameter of the circle. 2 2 (ii) Area = πr or πd /4 p=
(iii) Area of semi-circle =
pr 2 2
(iv) Area of a quadrant of a circle =
pr 2 4
Area enclosed by two concentric circles: If R and r are radii of two con-centric circles, then area enclosed by the two circles 2 2 2 2 = πR – πr = π (R – r ) = π (R + r) (R – r)
r
R
Some useful results: (i) If two circles touch internally, then the distance between their centres is equal to the difference of their radii. (ii) If two circles touch externally, then the distance between their centres is equal to the sum of their radii. (iii) Distance moved by a rotating wheel in one revolution is equal to the circumference of the wheel. (iv) The number of revolutions completed by a rotating wheel in one minute
=
Distance movedin oneminute Circumference
Arc, Chord, Segment, Sector of a Circle
Q M ajor S ector
M ajor S egm ent
O r A
M inor S egm ent
B
q
r
M inor S ecto r
A
B
P Arc: Any portion of circumference. e.g.: APB is minor arc while AQB is major arc. Chord: The line joining any two points on the circle. e.g.: AB. Segment: In figure chord AB divides the circle in two segments i.e., APBA (minor segment) and AQBA (major segment). Sector: The region bounded by the two radii AO and BO and arc AB is called sector of the circle. Let ∠AOB = ?, where ? is called central angle. Length of Arc: When sector angle ∠AOB = θ. We know that length of arc when sector angle (∠AOB = 360°) is 2πr length of arc when sector angle (∠AOB = 1°) is =
2 pr 360°
Length of arc when sector angle (∠AOB = θ) is =
2 pr ´ q 360°
Length of arc AB = 2pr ´ Perimeter of the sector = 2r ´
q 360°
q ´ 2pr 360°
Area of Sector: When sector angle ∠AOB = θ 2 We know that area of circle when sector angle (∠AOB = 360°) is πr pr 2 360°
Area of arc when sector angle (∠AOB = 1°) is = Area of arc when sector angle (∠AOB = θ) is =
pr 2 ´q 360°
q 360° Area of Segment (shaded) of A Circle : Area of Sector AOB – Area of triangle AOB 2 Area of Sector = pr ´
We can use formula given below when q ³ 90° pr 2 q q q Area of segment = - r 2 sin cos 360° 2 2
We can use formula given below when θ = 90° pr 2 q 1 2 Area of segment = - r sin q 360° 2
Perimeter of Segment (shaded) of A Circle: AB + arc (APB) Perimeter of Segment =
q 2prq - 2r sin 360° 2
