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.
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?
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).
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. . . .
Billy's class had a robot called Fred who could draw with chalk
held underneath him. What shapes did the pupils make Fred draw?
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?
Measure the two angles. What do you notice?
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.
How would you move the bands on the pegboard to alter these shapes?
Is it true that any convex hexagon will tessellate if it has a pair
of opposite sides that are equal, and three adjacent angles that
add up to 360 degrees?
Can you explain why it is impossible to construct this triangle?
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.
How can you make an angle of 60 degrees by folding a sheet of paper
Creating designs with squares - using the REPEAT command in LOGO.
This requires some careful thought on angles
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
This LOGO Challenge emphasises the idea of breaking down a problem
into smaller manageable parts. Working on squares and angles.
Can you find all the different triangles on these peg boards, and
find their angles?
More Logo for beginners. Learn to calculate exterior angles and draw regular polygons using procedures and variables.
Turn through bigger angles and draw stars with Logo.
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?
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?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.