Triangular Number Calculator
Find the nth triangular number — the count of dots in a triangle with n rows — with the formula and the running sum shown.
How it works
Tₙ = n(n + 1) ÷ 2 = 1 + 2 + 3 + … + n
The nth triangular number counts the dots you can arrange in an equilateral triangle with n dots along each side — 1, 3, 6, 10, 15 and so on. It equals the sum of the first n whole numbers, and the pairing trick (first + last, second + second-last, …) gives the tidy closed form n(n+1)/2. Triangular numbers appear in handshake counts, the diagonals of Pascal’s triangle and the classic "12 days of Christmas" totals.
Worked example
For n = 10, T₁₀ = 10 × 11 ÷ 2 = 55. That is exactly 1 + 2 + 3 + … + 10, and the next triangular number is T₁₁ = 11 × 12 ÷ 2 = 66.
Sources: Triangular number — Wikipedia
Frequently asked questions
What is a triangular number?
It is the total number of objects that form a filled triangle — one in the top row, two in the next, and so on down to n in the bottom row. The sequence runs 1, 3, 6, 10, 15, 21, …
How is it related to the sum of integers?
The nth triangular number is identical to 1 + 2 + … + n. Both use the formula n(n+1)/2, so a triangular number and the sum of the first n integers are the same thing.
What are triangular numbers used for?
They count pairwise relationships — the number of handshakes among n+1 people is Tₙ — and show up in Pascal’s triangle, staircase problems and cumulative totals.
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This calculator is for educational purposes. Double-check important results before acting on them.