Have a look at this very short video which illustrates Centred Shapes:

**Can you see how Centred Shapes are made?**

Now, take a look at this video which shows, using pictures, how centred triangles are related to triangular numbers and square numbers:

**Can you convince yourself that this will be the case for all Centred Triangles?**

If you would like to read more, take a look at this Word document.

Now take a look at some of these relationships and see what you can discover.

**Can create some pictures to show that the relationships are true (a visual proof)?**

**a/ 3 ^{rd} Centred Hexagon Number + 1 = 2 of 3^{rd} Centred Triangle Number**

(or N

^{th}Centred Hexagon Number + 1 = 2(N

^{th}Centred Triangle Number)

**b/ 3 ^{rd} Centred Square Number – 2^{nd} Triangular Number = 3^{rd} Centred Triangle Number**

(or N

^{th}Centred Square Number – (N-1)

^{th}Triangular Number = N

^{th}Centred Triangle Number)

**c/ 3² + 2² = 3 ^{rd} Centred Square Number**

(or N² + (N-1)² = N

^{th}Centred Square Number)

**d/ 2 ^{nd} Triangular Pyramid Number + 3^{rd} Triangular Pyramid Number = 3^{rd} Square Based Pyramid Number**

(or (N-1)

^{th}Triangular Pyramid Number + N

^{th}Triangular Pyramid Number = N

^{th}Square Based Pyramid Number)

Once you've had a go at these, you may like to look at Centred Shapes - More to Explore.