Question

use the formula for the circumference of a circle to write a formula for the area of a circle in terms of its circumference

Answer 1

The formula for the area of a circle in terms of its circumference is A = (C²) / (4π), by substituting the expression for the radius r = C / (2π) into the area formula A = πr².

Step-by-step explanation:

The student has asked how to use the formula for the circumference of a circle to write a formula for the area of a circle in terms of its circumference. The formula for the circumference (C) of a circle is C = 2πr, where π (pi) is approximately 3.14159 and r is the radius of the circle.

To write the formula for the area (A) in terms of the circumference, we first solve the circumference formula for r: r = C / (2π). Then we substitute this expression for r into the formula for the area of a circle, which is A = πr². This gives us A = π (C / (2π))², simplifying, we get A = (C²) / (4π).

Thus, the formula for the area of a circle in terms of the circumference is A = (C²) / (4π).

Answer 2

`A=(C^2)/(4\pi ^2)`

Step-by-step explanation:

Recall to find the circumference of a circle the formula is
`C=\pi d` or
`C=2\pi r`. We will also need the formula for the area of a circle which is
`A=\pi r^(2)`. Since the area formula is in terms of r we will use the second formula for circumference.

We start by solving for r in the Circumference formula.

`C=2\pi r\\(C)/(2\pi ) =r`. We input this value of r into the area formula.

`A=\pi r^(2) \\A=\pi ((C)/(2\pi ))^2\\A=\pi ((C^2)/(4\pi ^2 ))\\A=(C^2)/(4\pi ^2)`