The circumference of a circle is doubled. By what factor is its area increased?

The circumference of a circle can be found by multiplying the diameter (which is twice the length of the radius) by π - in other words, C = dπ = 2πr. Notice what happens when we double that:

2C = 2 (2πr)

It doesn’t seem like much, but shifting the terms around is revealing:

2C = 2π2r

When we double the circumference, * we also double the radius*

Now remember that the area of a circle is π time the circle’s radius squared - A = πr^2. What happens when we double the circumference? Well, since we know that doubling the circumference doubles the radius (r becomes 2r) we can find out what happens by to the area by substituting 2r for r:

A = π (2r) ^2 = π4r^2 = 4πr^2

When the circumference doubles, the area * quadruples*; the circumference increases by a factor of 2, while the area increases by a factor of 2 * squared*, or 4.