A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

This article gives an wonderful insight into students working on the Arclets problem that first appeared in the Sept 2002 edition of the NRICH website.

Take a look at the photos of tiles at a school in Gibraltar. What questions can you ask about them?

Draw some stars and measure the angles at their points. Can you find and prove a result about their sum?

Draw some angles inside a rectangle. What do you notice? Can you prove it?

Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.

Can you work out how these polygon pictures were drawn, and use that to figure out their angles?

Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

Drawing the right diagram can help you to prove a result about the angles in a line of squares.

Creating designs with squares - using the REPEAT command in LOGO. This requires some careful thought on angles

This LOGO Challenge emphasises the idea of breaking down a problem into smaller manageable parts. Working on squares and angles.

Are these statements always true, sometimes true or never true?

The graph below is an oblique coordinate system based on 60 degree angles. It was drawn on isometric paper. What kinds of triangles do these points form?

Can you find all the different triangles on these peg boards, and find their angles?

Can you explain why it is impossible to construct this triangle?

How can you make an angle of 60 degrees by folding a sheet of paper twice?

How would you move the bands on the pegboard to alter these shapes?

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

Turn through bigger angles and draw stars with Logo.

Billy's class had a robot called Fred who could draw with chalk held underneath him. What shapes did the pupils make Fred draw?

More Logo for beginners. Learn to calculate exterior angles and draw regular polygons using procedures and variables.

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?