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Degree Ceremony

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

Logosquares

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Geometric Trig

Age 16 to 18 Short Challenge Level:

There are many ways to create the solution as it is something like a jigsaw. We used this method
 
1. Mark the right angles (use the fact the a radius and tangent at a point are at right angles)
2. Make all occurrences of the angle a
3. Mark the unit length
 
The diagram then becomes
 
 
To work out all of the areas we need to decide what unit of measurement the angle $a$ is in. Of course, we choose radians: there are $2\pi$ radians in a circle. We will also need to know the formula for the area of a circle and the area of a triangle. In this case, the areas are given as


The two largest areas are equal for around 0.40523 radians (23.2 degrees).