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.
The three corners of a triangle are sitting on a circle. The angles
are called Angle A, Angle B and Angle C. The dot in the middle of
the circle shows the centre. The counter is measuring the size. . . .
Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).
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?
Can you work out how these polygon pictures were drawn, and use that to figure out their angles?
Measure the two angles. What do you notice?
Billy's class had a robot called Fred who could draw with chalk
held underneath him. What shapes did the pupils make Fred draw?
What can you say about the angles on opposite vertices of any
cyclic quadrilateral? Working on the building blocks will give you
insights that may help you to explain what is special about them.
Draw some stars and measure the angles at their points. Can you find and prove a result about their sum?
Show that among the interior angles of a convex polygon there
cannot be more than three acute 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
Which hexagons tessellate?
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?
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?
Are these statements always true, sometimes true or never true?
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
Turn through bigger angles and draw stars with Logo.
More Logo for beginners. Learn to calculate exterior angles and draw regular polygons using procedures and variables.
What is the relationship between the angle at the centre and the
angles at the circumference, for angles which stand on the same
arc? Can you prove it?
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
How would you move the bands on the pegboard to alter these shapes?
How many different triangles can you make which consist of the
centre point and two of the points on the edge? Can you work out
each of their angles?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
Can you find all the different triangles on these peg boards, and
find their angles?
This LOGO Challenge emphasises the idea of breaking down a problem
into smaller manageable parts. Working on squares and angles.