Math
Triangular Numbers Calculator
Triangular numbers count the dots in a triangle with n rows: 1, 3, 6, 10, 15, … This calculator gives the nth triangular number from n(n + 1)/2, checks whether a number is triangular by testing whether 8N + 1 is a perfect square (and finds which triangle it is), lists every triangular number in a range, and adds up the first n triangular numbers to get the tetrahedral number n(n + 1)(n + 2)/6.
Row count of the triangle.
Try an example
Result
Result
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- Formula used
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- Index n
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- Neighbouring triangular numbers
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- Also
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Study path
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What the triangular numbers calculator solves
A triangular number counts the dots in a triangle: one dot in the first row, two in the second, and so on. The nth triangular number is therefore 1 + 2 + … + n, which Gauss's pairing trick turns into n(n + 1)/2. The first few are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55. This page computes any T_n, decides whether a given number is triangular, lists them, and sums them.
| n | T_n | Dots | T_n + T_(n+1) |
|---|---|---|---|
| 1 | 1 | ● | 4 = 2^2 |
| 2 | 3 | ● / ●● | 9 = 3^2 |
| 3 | 6 | ● / ●● / ●●● | 16 = 4^2 |
| 4 | 10 | four rows | 25 = 5^2 |
| 5 | 15 | five rows | 36 = 6^2 |
| 10 | 55 | ten rows | 121 = 11^2 |
How to tell if a number is triangular
Solve n(n + 1)/2 = N for n with the quadratic formula: n = (-1 + √(1 + 8N))/2. So N is triangular exactly when 8N + 1 is a perfect square. For 5,050: 8 × 5,050 + 1 = 40,401 = 201^2, giving n = (201 - 1)/2 = 100. For 5,000: 40,001 is not a square (200^2 = 40,000), so 5,000 is not triangular; it sits between T_99 = 4,950 and T_100 = 5,050.
Sums of triangular numbers
Stacking triangles makes a tetrahedron: T_1 + T_2 + … + T_n = n(n + 1)(n + 2)/6, the nth tetrahedral number (1, 4, 10, 20, 35, …). And two consecutive triangular numbers always add to a perfect square, T_n + T_(n+1) = (n + 1)^2, which you can see by fitting two staircases together.
How to use it
- Choose a mode: nth term, check a number, list a range, or sum the first n.
- Enter n or the number and read the result with the formula substitution.
- Open Show the work to see n(n + 1)/2 evaluated, the 8N + 1 test, or the term-by-term sum.
How to read the answer
In nth-term mode the result is the dot count of an n-row triangle and equals 1 + 2 + … + n. In check mode, Yes comes with the index n so you know which triangle it is; No comes with the two triangular numbers on either side. The list mode shows the values with their The small raised number saying how many times to multiply the base by itself., and the sum mode returns the tetrahedral number, the total of all triangles up to n rows.
Common mistakes and edge cases
- Using n^2/2 or n(n - 1)/2 for T_n. The formula is n(n + 1)/2; n(n - 1)/2 is the previous triangular number T_(n-1).
- Forgetting to add 1 in the 8N + 1 test (8N alone is never a perfect square for N > 0 unless N is twice a square).
- Treating 0 as non-triangular; T_0 = 0 is usually included.
- Confusing triangular numbers with triangle-related geometry; they are a counting sequence.
- Reading the check result's index as the number itself: 5,050 is triangular with index 100, not 5,050.
Worked examples
100th triangular number
1 + 2 + … + 100 = 5,050.
Result
5,050
T_0
The empty triangle has 0 dots.
Result
0
Is 5050 triangular?
8 × 5050 + 1 = 201^2, n = 100.
Result
Yes
Is 5000 triangular?
40,001 is not a perfect square.
Result
No
Triangular numbers from 1 to 200
1, 3, 6, … , 190.
Result
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190
Sum of the first 10 triangular numbers
10 × 11 × 12 / 6 = 220.
Result
220
Large index
T_1000000 = 500,000,500,000.
Result
500,000,500,000
Negative number
Never triangular.
Result
Error
Frequently asked questions
What is a triangular number?+
A number that can be arranged as an equilateral triangle of dots: 1, 3, 6, 10, 15, … The nth one is the sum 1 + 2 + … + n = n(n + 1)/2.
What is the formula for the nth triangular number?+
T_n = n(n + 1)/2. For n = 7, T_7 = 7 × 8 / 2 = 28.
How do I check whether a number is triangular?+
Compute 8N + 1. If it is a perfect square k^2, then N is triangular with index n = (k - 1)/2. For N = 21: 8 × 21 + 1 = 169 = 13^2, so n = 6.
Is 0 a triangular number?+
Yes, by convention T_0 = 0 (the triangle with no rows). Some lists start at T_1 = 1.
What is the sum of the first n triangular numbers?+
n(n + 1)(n + 2)/6, the nth tetrahedral number. The first ten triangular numbers add to 220.
Why do two consecutive triangular numbers add to a square?+
Two staircases of n and n + 1 rows fit together into an (n + 1) × (n + 1) square: T_n + T_(n+1) = (n + 1)^2, e.g. 10 + 15 = 25.
Which numbers are both triangular and square?+
1, 36, 1225, 41616, … They come from solutions of a Pell equation; 36 = T_8 = 6^2.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026