Calculators
Tetrahedral Number Calculator
Calculate the nth tetrahedral number and sum of the first n triangular numbers exactly in your browser.
Sum triangular layers into a tetrahedron.
The nth tetrahedral number is Te(n) = n × (n + 1) × (n + 2) ÷ 6. It counts spheres stacked in triangular layers of sizes 1, 3, 6, 10, and so on. For n = 4, the sum is 1 + 3 + 6 + 10 = 20. For n = 10, the result is 220. We include Te(0) = 0 for no layers. Enter a non-negative index of up to 100 digits, without separators, signs, decimals or scientific notation. Leading zeros are accepted. Exact browser BigInt arithmetic avoids rounding even for large results. The tool calculates a tetrahedral number from a layer count; it does not test membership in the sequence. All processing stays in your browser.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is a tetrahedral number?
It counts objects stacked in a triangular pyramid with n triangular layers. The first five positive tetrahedral numbers are 1, 4, 10, 20, and 35.
How is the nth tetrahedral number calculated?
Multiply n by n + 1 and n + 2, then divide by 6. For 10 layers, 10 × 11 × 12 ÷ 6 = 220.
How does it relate to triangular numbers?
A triangular number counts one triangular arrangement. A tetrahedral number adds the first n triangular numbers. Four layers contain 1 + 3 + 6 + 10 = 20 objects.
Can I enter zero or fractional layers?
Zero returns zero for an empty stack. Negative and fractional indices are rejected. Use digits only, with no signs, commas or scientific notation.
Are large results rounded or uploaded?
No. Browser BigInt arithmetic gives exact results for indices up to 100 digits. Calculations run locally without uploading your input.