Question

A large circular dinner plate has a radius of

20 centimeters. A smaller circular plate with a
circumference of 20 centimeters is placed in
the center of the larger dinner plate. What is
the area of the part of the larger dinner plate
that is not covered by the smaller plate?

Answer 1

Approximately
`1225\; {\rm cm^(2)`.

Step-by-step explanation:

The smaller circular plate entirely overlap with the larger plate. Hence, the area of the larger plate that is uncovered would be equal to the difference between the area of the two plates. Given the radius of the larger plate and the circumference of the smaller plate, this area difference can be found in the following steps:

• Find the radius of the smaller plate from its circumference.
• Find the area of the two plates from the radius of each plate.
• Subtract the area of the smaller plate from that of the larger plate to find the difference in area.

If the radius of a circle is
`r`, the circumference of this circle would be
`C = 2\, \pi\, r`. Rearrange this equation to find the radius of this circle in terms of its circumference:

`\displaystyle r = (C)/(2\, \pi)`.

Given that the circumference of the smaller circular plate is
`C = 20\; {\rm cm}`, radius of this plate would be
`r = (20 / (2\, \pi))\; {\rm cm} = (10 / \pi)\; {\rm cm}`.

If the radius of a circle is
`r`, the area of that circle would be
`A = \pi\, r^(2)`.

• The area of the larger circular plate would be
  `(20})^(2)\, \pi\; {\rm cm^(2)} = 400\, \pi\; {\rm cm^(2)}`.
• The area of the smaller circular plate would be
  `(10 / \pi)^(2)\, \pi\; {\rm cm^(2) = (100 / \pi)\; {\rm cm^(2)}`.

The difference between the area of the two plates would be:

`\displaystyle (400\, \pi)\; {\rm cm^(2)} - \left((100)/(\pi)\right) \; {\rm cm^(2)} \approx 1225\; {\rm cm^(2)}`.