Visualising Triangular Numbers

Triangular numbers can be represented by a triangular array of squares.
Here is the sixth triangular number:

Imagine two copies of the triangular array above. Can you picture how to fit them together to make a rectangle?

What are the dimensions of your rectangle?

Can you write down the dimensions of the rectangle made from two copies of the 250th triangular number? Can you use this to work out the 250th triangular number?  

Can you deduce a strategy for working out any triangular number?

Here are some triangular number relationships. Can you use diagrams to prove them?

Add together two consecutive triangular numbers. The answer is a square number.
Multiply any triangular number by eight and add one. The answer is always a square number.
Multiply any triangular number by nine and add one. The answer is always another triangular number.

We'd love you to share how you've tried this and what you've discovered. You can add a comment below, or you can email us - your work may be featured in the showcase.

Comments

The dimensions of this rectangle are: 7 cm by 6cm. This means that the area is 42cm squared.

Well done! What does that mean for the dimensions of the rectangle made from the 250th triangular number? Would this help you to work out the 250th triangular number? Or any triangular number?

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