### Floored

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?

### Pie Cuts

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).

### Getting an Angle

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

# Triangles in Circles

### Why do this problem ?

This problem challenges students to apply what they know about angles in triangles. The nine-peg circle allows students to concentrate on the geometrical structure without having to worry about the arithmetic. It offers a good preparation for the problems Subtended angles and Right angles which lead towards the circle theorems.
Teachers may find the article Angle Measurement: an Opportunity for Equity of interest.

### Possible approach

All students will need a sheet of nine-peg circles to jot down ideas during discussion.

Demonstrate how the geoboard works - clicking on a coloured rubber band, dragging it onto a peg and then "stretching" it out onto two more pegs to make a triangle.
"How many different triangles can you make which consist of the centre point and two of the points on the edge? "
"Can you find all the angles of the triangles?"
At some stage students will want to know what makes triangles "different". For the purposes of this exercise, triangles that are congruent are considered to be the same.
After a short time bring the class together to discuss findings. Issues that you could touch upon:
• caclulating angles at the centre for different numbers of pegs
• rules for finding the base angles of the isosceles triangles
Draw a triangle on the geoboard with three vertices on the edge of the circle which encloses the centre point.
"Can you find all the angles of this triangle?"
How this builds on their previous work may not be immediately obvious to most students. After a few minutes, if no progress is being made that can be shared, add one isosceles triangle inside the original triangle (see the solution ). A further isosceles triangle can be added if needed.
Discuss efficient methods.
"Now draw all the possible triangles with three vertices on the edge of the circle and find their angles."
At some stage discuss how to cope with triangles that do not enclose the centre.

This handout presents the whole problem on one sheet.

### Key questions

What do we know already that might be useful here?

### Possible extension

The virtual geoboard environment allows you to create triangles in circles with a variable number of pegs. The virtual geoboard resource also has links to sheets with printed circles with different numbers of pegs, a full set of these sheets can also be found here.
Ask students to work out the angles of triangles in 10-peg, 12-peg, 15-peg ... circles.

Lens Angle provides an interesting challenge that requires students to apply the properties of triangles in circles.

### Possible support

Before moving on to the main activity, students find the angles of triangles with a vertex at the centre in circles with different number of pegs on the edge.

Students can start by finding the angles of triangles in 9-peg and 12-peg circles which contain the centre of the circle. They can then move on to triangles that do not contain the centre of the circles.
Printable sheets with all the possible triangles in 9-peg and 12-peg circles can be found here and here .
Sethe extension section of these notes for other useful links.