How this instrument works
A triangular number counts the dots in a triangular stack: row 1 has a single dot, row 2 adds two more, row 3 adds three more, and so on, with the nth triangular number Tₙ being the running total after n rows. Written out, Tₙ = 1 + 2 + 3 + ... + n, and rather than adding every term one at a time, the closed-form shortcut Tₙ = n(n+1) ⁄ 2 gives the same total instantly.
That shortcut is famously credited to a young Carl Friedrich Gauss, who is said to have found it by pairing the first and last terms of the sum: 1 with n, 2 with (n−1), and so on, each pair adding to the same total of (n+1). There are n ⁄ 2 such pairs, so the whole sum is n ⁄ 2 copies of (n+1) — exactly n(n+1) ⁄ 2, derived without adding a single number by hand.
Triangular numbers show up well beyond dot-stacking: the number of handshakes possible in a room of n+1 people, the number of games in a single round-robin tournament with n+1 teams, and the count of edges in a complete graph all reduce to the same formula, since each of those situations is really counting every unique PAIR drawn from a set — a structure identical to summing 1 through n.
- Enter the row count into the n field — how many rows deep the triangular stack goes.
- Read nth triangular number: the sheet computes n(n+1) ⁄ 2 directly, without summing term by term.
- Increase n to see how quickly the total grows — triangular numbers grow roughly with the square of n, not linearly.
Worked example — a 6-row triangle
A triangular stack six rows deep has T₆ = 6 × 7 ⁄ 2 = 21 dots total — the same answer as adding 1 + 2 + 3 + 4 + 5 + 6 by hand, but reached with a single multiplication and division instead of five separate additions.
Compare a 10-row stack: T₁₀ = 10 × 11 ⁄ 2 = 55. Growing the row count from 6 to 10, an increase of less than double, more than doubled the dot total — a reminder that triangular numbers grow roughly with the square of n, since the formula's numerator carries an n² term once expanded.
Questions
What is the formula for the nth triangular number?
Tₙ = n(n+1) ⁄ 2. It equals the sum 1 + 2 + 3 + ... + n computed all at once, via the classic pairing trick of matching the first term with the last, the second with the second-to-last, and so on, each pair summing to the same value.
Why is it called a 'triangular' number?
Because the running totals correspond exactly to the dot counts of triangles built from stacked rows — 1, then 1+2=3, then 1+2+3=6, then 1+2+3+4=10 — each total forming a slightly larger triangular arrangement of dots than the one before it.
Is there a connection between triangular numbers and combinations?
Yes — the nth triangular number equals the number of ways to choose 2 items from a set of n+1, written C(n+1, 2). This is why triangular numbers count things like handshakes in a room or games in a round-robin tournament: both are really counting unique pairs.
What is the first triangular number?
T₁ = 1, a single dot — the base case the whole sequence builds from. T₀ = 0 is also sometimes included as a trivial starting point, representing an empty stack with no rows at all.
How fast do triangular numbers grow?
Roughly with the square of n, since expanding n(n+1) ⁄ 2 gives (n² + n) ⁄ 2, and the n² term dominates for larger n. Doubling n roughly quadruples Tₙ, a much faster growth rate than the plain, linear row count itself.