## Total Surface Base Area of Nested Pizza Pans

Suppose we have a stack of pizza pans. The pans are stacked so that one pan fits snugly inside the next smallest one, up to the largest, which has an inside radius of 50cm. The rim of each pan has a thickness of 1cm
We find

\begin{aligned} & \pi\times 50^2 + \pi \times 49^2 + ... + \pi \times 1^2 \\ &= \pi (50^2 + 49^2 +... +1^2 ) \end{aligned}

We can use the formula for the sum of
$n$
square numbers.
$S_n = \frac{n}{6}(n+1)(2n+1)$

We have
$S_{50} =\frac{50}{6}(50+1)(2 \times 50+1)=42925$
.
The total base area of the pans is then
$42925 \pi$
cm2 .