To find the diameter and the perimeter of a circular table when given the area, we need to use the formulas related to the geometry of a circle.
a. Find the diameter of the circular table:
The area of a circle (A) is given by the formula:
[tex]\[ A = \pi r^2 \][/tex]
where [tex]\( r \)[/tex] is the radius of the circle, and [tex]\( \pi \)[/tex] (Pi) is a mathematical constant approximately equal to 3.14159.
From the area equation, we can solve for the radius [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{A}{\pi}} \][/tex]
Given the area of the circular table is 22,464 [tex]\( \text{cm}^2 \)[/tex], we substitute this value back into the equation to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{22464}{\pi}} \][/tex]
After calculating the radius, we can find the diameter ([tex]\( d \)[/tex]) of the circular table. The diameter is twice the radius:
[tex]\[ d = 2r \][/tex]
Plugging in the values for the radius, we find that the diameter of the circular table is approximately 169.12 cm.
b. Find the perimeter (circumference) of the circular table:
The perimeter (or circumference) of a circle (C) is given by the formula:
[tex]\[ C = 2\pi r \][/tex]
Using the previously calculated radius, we can find the perimeter by inserting the radius into the circumference formula:
[tex]\[ C = 2\pi r \][/tex]
After substituting the value of the radius into the equation, the calculated perimeter (or circumference) of the circular table is approximately 531.31 cm.
Therefore, the diameter of the circular table is 169.12 cm, and the perimeter is 531.31 cm.
