Triangular Number Calculator

How to use the Triangular Number Calculator

1. 1Enter the number n, or the number to test.
2. 2Pick the units or options from the menus if the triangular numbers tool offers them.
3. 3Read the triangular numbers in the large result box.
4. 4Look at the extra rows for related figures.
5. 5Press Copy result to keep the answer.

When you need the triangular numbers

Teachers use triangular numbers to introduce sequences and sums. They also count handshakes at a party: n people shake hands in T(n − 1) ways. Programmers use them for loop counts, and puzzle fans meet them in bowling pins and billiard racks.

Triangular numbers are linked to many others. The sum of two consecutive triangular numbers is a square, and every even perfect number is triangular. They also appear in the third diagonal of Pascal's triangle.

How the triangular numbers is worked out

The nth triangular number is n × (n + 1) ÷ 2. A number T is triangular if 8T + 1 is a perfect square; then n = (√(8T + 1) − 1) ÷ 2. The 10th triangular number is 55 and the 100th is 5,050, the sum that the young Gauss is said to have found in seconds. The number 5,050 is triangular with n = 100, while 56 is not. The first twelve are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 and 78.

Every answer on this page comes from that one rule, so you can check any result by hand with a calculator. The figures shown underneath, such as other units, are conversions of the same answer.

• Nth number and a test
• Very large numbers
• First twelve listed

Mistakes to avoid with triangular numbers

The calculator accepts digits only, up to 30 of them. When testing a number, enter the number itself, not its position. The position n is one more than the number of rows in the triangle if you count from 1, which matches the usual definition.

On phone and computer

On a phone, type into the boxes one after another and read the triangular numbers underneath; Copy result puts it on the clipboard for a message or a note. On a computer the inputs sit on the left and the answer on the right, so you can change one figure and watch the triangular numbers move. The triangular number calculator works the same on both. Triangular numbers sit in several patterns. Compare them with the Number Sequence Calculator, see rows in Pascal's Triangle Generator, and check Fibonacci terms with the Fibonacci Calculator.

Features

• Gauss pairs
• 8T + 1 square test
• Handshake counts
• Clear messages when the triangular numbers inputs make no sense
• Copy the result with one tap
• Private: nothing is uploaded

Frequently asked questions