Question

In a circle, area and circumference can be found using the formulas A=(pi)r^2 and C=2(pi)r, respectively, Write a formula for A in terms of C.(your answer should not contain r)

Answer 1

Answer:

$A = \dfrac{C^2}{4\pi}$

Step-by-step explanation:

Given that

Area of a circle $A=\pi r^2$

Circumference of the circle $C = 2 \pi r$

Let us re-write the equation of area of circle:

$A=\pi r^2$

$A = \pi \times r \times r$

Multiplying and dividing with 2:

$A = \dfrac{2\pi r}{2} \times r\\\text{Putting }2\pi r = C:\\\Rightarrow A = \dfrac{C}{2} \times r\\\text{Multiplying and dividing by } 2\pi:\\\Rightarrow A = \dfrac{C}{2} \times \dfrac{2\pi r}{2\pi}\\\text{Putting }2\pi r = C:\\\Rightarrow A = \dfrac{C \times C}{4 \pi}\\\Rightarrow A = \dfrac{C^2}{4 \pi}$

Hence, A in terms of C can be represented as:

$A = \dfrac{C^2}{4\pi}$