If the radius of a circle is doubled, by what factor does the area of the circle

If the radius of a circle is doubled, by what factor does the area of the circle increase?

If the radius of a circle is doubled, by what factor does the area of the circle increase?

Step 1: Understand the formula for the area of a circle, which is A = πr², where r is the radius.
Step 2: Identify what happens when the radius is doubled. If the original radius is r, the new radius will be r' = 2r.
Step 3: Substitute the new radius into the area formula. The new area A' will be A' = π(2r)².
Step 4: Calculate (2r)². This equals 4r².
Step 5: Substitute this back into the area formula: A' = π(4r²) = 4πr².
Step 6: Compare the new area A' with the original area A. The original area A = πr².
Step 7: To find the factor of increase, divide the new area A' by the original area A: (4πr²) / (πr²) = 4.
Step 8: Conclude that the area increases by a factor of 4 when the radius is doubled.

Area of a Circle – Understanding the formula A = πr² and how changes in the radius affect the area.
Scaling Factors – Recognizing how doubling the radius leads to a squared increase in area.