A triangular number is a number that can be arranged as a filled triangle of evenly spaced dots — one dot in the first row, two in the second, three in the third, and so on, with each row one dot longer than the last. The triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55...
**The pattern.** Each triangular number is the sum of consecutive whole numbers starting from 1. 1 = 1. 3 = 1+2. 6 = 1+2+3. 10 = 1+2+3+4. 15 = 1+2+3+4+5. The n-th triangular number is the sum of every whole number from 1 to n.
**The formula.** Rather than adding up every number from 1 to n by hand, the n-th triangular number can be found directly: T(n) = n(n+1) ÷ 2. This shortcut is attributed to a famous (if possibly apocryphal) story about a young Carl Friedrich Gauss, who is said to have quickly summed 1 through 100 in grade school by pairing the first and last numbers (1+100=101), the second and second-to-last (2+99=101), and so on — fifty pairs each summing to 101, giving 50×101=5,050 almost instantly, rather than adding all 100 numbers one at a time.
**Worked example: find the 8th triangular number.** T(8) = 8×9÷2 = 72÷2 = 36. Checking by direct addition: 1+2+3+4+5+6+7+8 = 36. Matches.
**Worked example: find the 12th triangular number.** T(12) = 12×13÷2 = 156÷2 = 78.
**Checking whether a given number is triangular.** Rearrange the formula to solve for n: if T = n(n+1)/2, then n = (−1 + √(1+8T)) ÷ 2. If this comes out to a whole number, T is triangular; if not, it isn't.
**Worked example: is 45 a triangular number?** n = (−1 + √(1+8×45)) ÷ 2 = (−1 + √361) ÷ 2 = (−1 + 19) ÷ 2 = 18 ÷ 2 = 9. Since 9 is a whole number, 45 is triangular — specifically, it's the 9th triangular number.
**Worked example: is 40 a triangular number?** n = (−1 + √(1+8×40)) ÷ 2 = (−1 + √321) ÷ 2 ≈ (−1 + 17.92) ÷ 2 ≈ 8.46. Not a whole number, so 40 is not triangular.
**A genuinely useful, practical application.** Triangular numbers describe the total number of handshakes possible in a group, or games in a round-robin tournament where everyone plays everyone else exactly once: with n people, the total number of unique pairings is T(n−1) = (n−1)n/2. For 10 people, that's T(9) = 9×10/2 = 45 unique handshakes or matches. This same formula also describes the total number of dots in a bowling pin triangle (10 pins = T(4), 4 rows) or a billiards rack (15 balls = T(5), 5 rows).
**The connection to perfect squares.** Adding two consecutive triangular numbers always produces a perfect square: T(3)+T(4) = 6+10 = 16 = 4². T(4)+T(5) = 10+15 = 25 = 5². This isn't a coincidence for a couple of specific examples — it holds for every consecutive pair, and it has a clean geometric explanation: two triangular arrangements of dots, one slightly larger than the other, fit together perfectly to tile a square.
**Common mistakes.** The most common error is misremembering the formula's structure — writing n²/2 instead of the correct n(n+1)/2, which gives a meaningfully different (and wrong) result for anything beyond n=1. A second common mistake, when checking whether a number is triangular, is stopping at an approximate decimal result and rounding it to the nearest whole number rather than correctly concluding the number isn't triangular at all — the check requires an *exact* whole number, not merely a close one.
**Worked example: verify T(15) two different ways.** Formula: T(15) = 15×16÷2 = 240÷2 = 120. Direct sum: 1+2+3+4+5+6+7+8+9+10+11+12+13+14+15. Pairing first-and-last (Gauss's trick): (1+15)+(2+14)+(3+13)+(4+12)+(5+11)+(6+10)+(7+9)+8 = 16×7 + 8 = 112+8=120. Both methods agree.
**Square triangular numbers, a rarer overlap worth knowing about.** A handful of numbers are both triangular and perfect squares at once — 1, 36, 1225, and 41616 are the first few. 36 is T(8) (8×9÷2=36) and also 6². These "square triangular numbers" get rare fast, governed by a more advanced recurrence relation, but they're a genuinely interesting intersection between two of this guide's core concepts (perfect squares and triangular numbers).
**Worked example applying the round-robin tournament formula.** A chess club has 15 members and wants everyone to play everyone else exactly once. Total games needed = T(14) = 14×15÷2 = 105 games — using n−1 rather than n directly in the formula, since each person doesn't play themselves.
**Worked example: find which triangular number is closest to, but not over, 500 — a "search" variation on the formula.** Solving n(n+1)/2 ≤ 500 approximately: n²≈1000, so n≈31.6, try n=31: T(31)=31×32÷2=496. Try n=32: T(32)=32×33÷2=528, which exceeds 500. So T(31)=496 is the largest triangular number not exceeding 500 — and, notably, 496 is also a perfect number, a coincidental overlap between two of this site's featured number classifications worth noting for its own curiosity value.
**Worked example: find the sum of the first 20 whole numbers using the formula, rather than adding them one at a time.** T(20) = 20×21÷2 = 420÷2 = 210. This same shortcut — sum of 1 through n equals n(n+1)/2 — applies to any "add up consecutive whole numbers starting from 1" problem, regardless of how large n gets, without ever needing to perform the full addition by hand.
**Worked example: is 78 a triangular number?** n = (−1+√(1+8×78))÷2 = (−1+√625)÷2 = (−1+25)÷2 = 24÷2=12. A whole number, so 78 is triangular — specifically T(12), matching the earlier worked example that directly computed T(12)=78 from the formula.
**Pentagonal and hexagonal numbers, briefly, for context on how this generalizes.** Triangular numbers are the simplest case of a broader family of "polygonal numbers" — square numbers (n²), pentagonal numbers, hexagonal numbers, and so on, each built from a different-sided regular polygon of dots rather than a triangle. Each family has its own formula, generalizing the same underlying idea of counting dots arranged in an expanding geometric shape.
For a related, differently-patterned sequence of famous numbers, see The Fibonacci Sequence, Explained.