Circles: Area and Perimeter

Synopsis

1. The distance around the boundary of the circle is called the perimeter or the circumference of the circle.
   
   
2. Circumference (perimeter) of a circle = πd or 2πr, where d is he diameter, r is the radius of the circle and .
    
3. Perimeter of a semi circle or protractor = πr + 2r
   
   
4. Perimeter of a quadrant =  
    
5. Distance moved by a wheel in 1 revolution = Circumference of the wheel.
   Number of revolutions in one minute = 
    
6. The region enclosed inside a circle is called its area.
    
7. Area of a circle = πr²
    
8. Area of a semi-circle = 
    
9. Area of a quadrant =
   
   
10. Circles having the same centre but different radii are called concentric circles.
    Area enclosed by two concentric circles = πR² - πr² = π(R² - r²) = π(R + r)(R - r)
    Where, R and r are radii of two concentric circles
    
    
11. The part of the circumference between the two end points of the chord is called an arc. In the figure, arc  is shown.
    
12. A diameter of circle divides a circle into two equal arcs, each known as a semi-circle.
    
    
13. An arc of a circle whose length is less than that of a semicircle of the same circle is called a minor arc.
    
    
14. An arc of a circle whose length is greater than that of a semicircle of the same circle is called a major arc.
    
    
15. Length of an arc = 
    
    
16. The region bounded by an arc of a circle and two radii at its end points is called a sector.
    If the central angle of a sector is more than , then the sector is called a major sector and if the central angle is less than , then the sector is called a minor sector.
    
17. Perimeter of sector of angle θ = 
    
    
18. Area of a sector of angle θ = 
19. Area of major sector = πr² – Area of minor sector
     
20. A chord divides the interior of a circle into two parts, each called a segment.
    The segment which is smaller than the portion of semi-circle is called the minor segment and the segment which is larger than the portion of semi-circle is called the major segment. In the circle shown, the yellow portion is the minor segment while the non-shaded portion is the major segment.
    
21. Perimeter of segment of angle θ = 
22. Area of minor segment = Area of sector - Area of ΔABC
    
23. Area of minor segment can also be written as:
    
     
24. Area of major segment = Area of the circle – Area of minor segment