Question

Find the area of a ring whose outer and inner radii are 19 cm and 16 cm respectively.
A. $330{\text{ cm}}^2$
B. $310{\text{ cm}}^2$
C. $320{\text{ cm}}^2$
D. $350{\text{ cm}}^2$

Answer 1

Hint: A ring is the space between two concentric (same center) circles.
Area of the ring = Area of the outer circle - Area of the inner circle.
Area of a circle of radius r units is $\pi r^2\text{ sq. units}$ .

Complete step-by-step answer:
Let's say that the radius of the outer circle is $r_1=19\text{ cm}$ and the radius of the inner circle is $r_2=16\text{ cm}$ .

The area of the ring will be given by:
Area of the ring = Area of the outer circle - Area of the inner circle.
Using the formula for the area of a circle, we get:
= $\pi r_1^2-\pi r_2^2$
Taking out $\pi$ as the common factor:
= $\pi (r_1^2-r_2^2)$
= ${\displaystyle \frac{22}{7}}({19}^2-{16}^2)$
Using the identity $a^2-b^2=(a+b)(a-b)$ , we get:
= ${\displaystyle \frac{22}{7}}(19+16)(19-16)$
= ${\displaystyle \frac{22}{7}}\times 35\times 3$
= $22\times 5\times 3$
= $330{\text{ cm}}^2$
Therefore, the correct answer option is A. $330{\text{ cm}}^2$ .
Note: The total circumference of the ring will be $2\pi (r_1+r_2)$ .
Similar techniques can be used for finding the area of other shapes. e.g. Area of a pathway around a rectangular park.
A similar concept is used to calculate the volume of the material needed to construct a hollow solid.