Area of a Circular Ring or Annulus


A circular ring (annulus) is plane figure bounded by the circumference of two concentric circles of two different radii. The area of a circular ring is found by subtraction the area of small circle from that of the large circle. The examples of annulus are the area of a washer and the area of a concrete pipe.

If A and a, R and r stands for the areas and the radii of two circles and {A_r} for the area of the ring, the

{A_r} = A - a = \pi {R^2}  - \pi {r^2} = \pi ({R^2} - {r^2}) = \pi (R - r)(R + r)


i.e. to find the area of a ring (or annulus), multiply the product of the sum and the difference of the two radii by \pi in first figure.
Note: Rule holds good even when circles are not concentric as in second figure.


A path 14cm wide, surrounds a circular lawn whose diameter is 360cm. Find the area of the path.


Given that
            Radius of inner circle  = 180cm
            Radius of outer circle  = (180 + 14) = 194cm
            \therefore Area of path  = \pi (R - r)(R + r)
                                       =  \frac{{22}}{7}(194 + 180)(194 - 180) = 16456 Square cm


The areas of two concentric circles are 1386square cm and 1886.5 square cm respectively. Find the width of the ring.


Let R and r be the radii of the outer and inner circles respectively. Let d be the width of the ring then d = (R - r)


            \therefore Area of the outer circle  = \pi {R^2} = 1886.5 square cm
                                             or  R = 24.51 cm
            \therefore Area of the inner circle  = \pi {r^2} = 1386 square cm
            \therefore \log \pi +  2\log r = \log 1386
                                2\log  r = \log 1386 - \log (3.143)
                                              = 3.1418 - 0.4973 = 2.6454
                                  \log  r = 1.3223
                                          r =  anti\log (1.3223) = 21
            Hence, width of ring  = R - r = 24.51 - 21 = 3.51cm