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Degree Ceremony

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

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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.

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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.

The Dodecahedron

Age 16 to 18 Challenge Level:

How do you find the shortest distance between two points on the net of the dodecahedron?

 

Does the distance on the surface change when the flat net is folded into the shape of the dodecahedron?