Math 402

Journal assignment on triangle sums

Use your masking tape or rubber bands to make triangles on a sphere. If you have a big ball, use it. If not, use your tennis ball. For each triangle, calculate the sum of the interior angles. This is fairly easy to do even if you don't have a protractor. Draw a point P on the overhead. Put the point P at one vertex and draw that angle. Now move the overhead so that P lines up with a second vertex and draw that angle adjacent to the first (so that they share one ray). Now repeat with the last vertex.

What is the sum of the interior angles? (Give your best estimate.) Is it less than 180, greater than 180, or equal to 180?

Repeat this for several other triangles. Make some large and some small, with various types of angles.

Do you notice any pattern? Is the angle sum related to the size of the triangle?

Repeat this process for the hyperbolic space. Be careful not to destroy your space! If you use tape, you may want to make it less sticky first. Alternatively, try using ribbons or stips of paper.

Is the angle sum of each triangle less than 180, greater than 180, or equal to 180? Is the angle sum related to the size of the triangle?

Finally, repeat this process on the plane.

Again, is the angle sum of each triangle less than 180, greater than 180, or equal to 180? Is the angle sum related to the size of the triangle?