Question

A circle has a radius of 5 cm. If the area is tripled, what happens to the circumference?

Please give me a step-by-step explanation...

Answer 1

• The circumference is multiplied by √3

Explanation:

You can use the formulas for area and circumference to write the circumference in terms of area:

A = πr²

r = √(A/π) . . . . . solving for r

and

C = 2πr

C = 2π√(A/π) = 2√(Aπ)

Now, if A is multiplied by 3, we can use 3A in place of the A in this formula:

C' = 2√(3Aπ)

and we can factor out the original circumference, giving us ...

C' = (√3)(2√(Aπ))

C' = (√3)C . . . . the circumference is multiplied by √3

_____

You can work through these length/area/volume problems involving similar figures like this whenever you need to. I find it easier to remember that the scale factor is related to the power of the units.

Circumference is measured in units.

Area is measured in units².

So, the scale factor for area is the square of the scale factor for circumference. Or, the scale factor for circumference is the square root of the scale factor for area, which is what this problem is about.