finding circumference and area in terms of A and C?
The radius of a circle with an area ( A ) and circumference ( C ) is doubled.
A) How do I find the new area of the circle in terms of ( A )?
B) How do I find the new circumference of the circle in terms of ( C )?
I’m unsure how to approach this problem since I’m not accustomed to working without specific numerical values!
2 Answers
Hi, CG.
We’ll call the old radius r and the new radius r’ (“r-prime”).
We’re told that the radius is doubled. Thus, r’ = 2r (2 times r).
A) The area of a circle is π × (radius)²
So, the area of the old circle, A = πr²
Use the same formula to get the area of the new circle (let’s call it A’)
A’ = π × (r’)² = π × (2r)² = π × 2² × r² = 4 × πr²
Looking at the expression for A, we can substitute into the expression for A’ and get:
A’ = 4A. Thus the new area of the circle is 4 times the old area.
B) Circumference of a circle is 2π × (radius)
Thus, the old circumference is:
C = 2πr
And the new circumference is:
C’ = 2πr’ = 2π × (2r) = 4πr = 2 × 2πr
Substitute in the equation for C:
C’ = 2C
Thus, the new circumference is 2 times the old circumference
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