The area of a circle is 40 if the circle is dilated by a scale factor of 3 how much does the area grow?

Answer 1

Hi Nathaniel F.,

For Scale Factors (S.F.) of geometric figures:

Linear parts (one dimensional) will grow by multiplying by the (S.F.)¹.

Areas (two dimensional) will grow by multiplying by the (S.F.)².

Volumes (three dimensional) will grow by multiplying by the (S.F.)³.

So the area will grow by (S.F.)² or (3)² = 9

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Or, the radius of the given circle with area A₁ = 40:

40 = π*(r₁)²

(r₁)² = 40/π

r₁ = √(40/π)

The scaled up circle will have a radius 3*r₁ = r₂:

3*r₁ = 3*√(40/π) = r₂

The area (A₂) of the scaled up circle:

A₂ = π*[3*√(40/π)]² = 360

The area has a scale factor of A₂/A₁:

A₂/A₁ = 360/40 = 9 or A₂ = 9*A₁

I hope this helps, Joe.