Sum of Squares Calculator

About this calculator

The sum of the first n perfect squares, 1² + 2² + 3² + … + n², has a closed-form shortcut known since antiquity: n(n+1)(2n+1) ÷ 6, which gives the exact total without adding every term one by one. It is one of the classic results proved by mathematical induction in introductory algebra and number theory courses, alongside the simpler sum-of-first-n-integers formula.

Beyond being a standard proof exercise, this closed form appears in statistics as part of deriving variance and sum-of-squared-deviations formulas, in physics when summing discrete quantized energy levels, and in computer science as a textbook example for analyzing algorithm time complexity, since a nested loop that runs n² operations per outer iteration accumulates a total step count following exactly this formula.

Enter a value for n and this calculator returns the sum of squares from 1² through n².