You may also like

problem icon

Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

problem icon


Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

problem icon


Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Triangles Within Triangles

Stage: 4 Challenge Level: Challenge Level:1

This diagram shows how the first triangular number can be added to 3 copies of the second triangular number to make the fourth triangular number:

A triangle made from three triangular numbers plus one other

That is: $$ T_1 + 3 \times T_2 = T_4 $$ Here is a diagram showing how the second and third triangular numbers can be combined to make the sixth triangular number:

A triangular numebr made from three lots ofone triangular number plus another

$$ T_2 + 3 \times T_3 = T_6 $$ Can you generalise this rule?

Can you find a rule in terms of $ T_n $ and $T_{n+1}$?