Area of a Circular Ring or Annulus

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 subtracting the area of the small circle from that of the large circle. An example of an annulus is the area of a washer and the area of a concrete pipe.

If $$A$$ and $$a$$, $$R$$ and $$r$$ stand 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 the first figure.

Note: Rule holds good even when circles are not concentric as in second figure.

 

Example:

A path $$14$$cm wide surrounds a circular lawn with a diameter of $$360$$cm. Find the area of the path.

 

Solution:

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

 

Example:

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

 

Solution:

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

            $$\therefore $$ the area of the outer circle $$ = \pi {R^2} = 1886.5$$ square cm
or  $$R = 24.51$$ cm
$$\therefore $$ the 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, the width of the ring $$ = R – r = 24.51 – 21 = 3.51$$cm