Centred Shapes - More to Explore

Once you have got to grips with Centred Shapes, have a look for relationships between different centred shapes, triangular numbers and square numbers.
If you think you have found a relationship, try to find a diagram to explain why it works.

Here are some relationships you could try to prove:

• 6 x N^th Triangular Number + 1 = N^th Centred Hexagon 
• N² + (2N – 2)^th Triangular Number = N^th Centred Hexagon
• 3(N-1)^th Triangular Number + N^th Centred Triangular Number = N^th Centred Hexagon
• N(N+1) + N² = (2N)^th Triangular Number
• N^th Centred Pentagon = N^th Triangular Number + (2N-2)^th Triangular Number
• N^th Centred Pentagon – N² + N = N^th Centred Triangle
• N^th Triangular Pyramid - N² =  (N-2)^th Triangular Pyramid

For some further challenges, have a look at More Centred Shape Relationships.