Subject: Angles
Name: Rayna
Who is asking: Student
Level: Secondary

I am doing a presentation report on angles which has to be fun and entertaining as well as educational. I am having problems locating resources on angles that give me ideas of fun entertaining projects. My lecture is to be about 20 minutes long infront of a class of 11th and 12th graders. Please Help if you can.


Hi Rayna,

I have three suggestions. One from Claude:

How about talking about pool? There are a lot of sites in the games sections of browsers (at least yahoo), and some of them talk about the mathematical aspects, like the angles and so on.

and two from Walter:

My suggestions for interesting 'angle' projects would be to move one from the plane to look at angles in other situations. Here are TWO possibilities:

  1. Angles on the Sphere.

    The sum of the angles of a triangle on a sphere is always >= 180 degrees. Infact, if you know this sum you can compute the area (sum 180 degrees means the area is 0!). Moreover, if two traingles have the SAME three angles, then they are congruent: AAA congruence - rather like SSS congrunce in the plane.

    You could probably prove the angle sum / area formula for the sphere, within a project. A possible source is:

    This is a draft textbook for first year university geometry - but only assumes about grade 11/12 math (no calculus or linear algebra).

  2. Angles in polyhedra.

    Take the regular polyhedra. Add up the angles in all the faces. Now consider adding up the 'angle deficits': At each vertex find out how much LESS than 360 (a flat full circle) the sum of the angles in the faces is. Add up these deficits.

    You should find that the sum of these deficits is the SAME for all the regular polyhedra. Consider other non-regular polyhdra. Find the same angle deficit sum.

    There is a theorem lurking in this constant. It is related to 'Euler's Formula'. (What we are doing is actually older than Euler - it goes back to Descartes.)

    A search on the internet under Euler's Formula should give some information on these results.

Hope those suggestions are of some interest. Angles appear lots of places with lots of uses. Plane geometry is just the simplest starting point - not the end of the story.

The Centralizer

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