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Algebra

Sum of Squares

n(n+1)(2n+1)/6. Free online Sum of Squares. Calculate sum of squares online — fast, accurate, mobile-friendly, no signup needed.

Sum
385

Derivation

  1. ├── 01Givenn = 10
  2. ├── 02Formulat × (t+1) × (2 × t+1) / 6
  3. └── 03Compute Sum385
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§01What is

Understanding the Sum of Squares

The Sum of Squares computes Sum from 1 input: n. n(n+1)(2n+1)/6.

Algebra is the art of solving for the unknown. Rearranging a formula to isolate the variable you actually need is the single most common real-world math skill — and doing it with real numbers under time pressure is where errors happen. The Sum of Squares sits in that toolkit — it n(n+1)(2n+1)/6. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

t × (t+1) × (2 × t+1) / 6

Where

n
N
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: N = 10.

  1. 01Start by noting the input — N: 10.
  2. 02Substitute these values into the formula: t × (t+1) × (2 × t+1) / 6
  3. 03Compute Sum: the calculator returns 385.
  4. 04Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Sum of Squares Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

N halved

n = 5 (from 10)

Keep every other input at its default and halve the n. See how sum responds.

  1. 01New N: 5
  2. 02Baseline Sum: 385
  3. 03New Sum: 55
  4. 04Sum decreases by 85.7% → use this sensitivity to plan for real-world variation.
02 · PATTERN

N doubled

n = 20 (from 10)

Keep every other input at its default and double the n. See how sum responds.

  1. 01New N: 20
  2. 02Baseline Sum: 385
  3. 03New Sum: 2870
  4. 04Sum increases by 645.5% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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