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.

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

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

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

Make five different quadrilaterals on a nine-point pegboard, without using the centre peg. Work out the angles in each quadrilateral you make. Now, what other relationships you can see?

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

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and. . . .

Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.

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.

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

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

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

The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.

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

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

Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST and PU are perpendicular to AB produced. Show that ST + PU = AB

Join pentagons together edge to edge. Will they form a ring?

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

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE.

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

Turn through bigger angles and draw stars with Logo.

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

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

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

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

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?