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# Platonic Planet

Glarsynost the alien lives on a platonic planet whose shape is that of a perfect regular dodecahedron.

Every day, she likes to go for a walk to have a look at her planet and see if she has any visitors.

What fraction of the planet's surface can she see from the middle of a face? From an edge? From a vertex?

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Age 14 to 16

Challenge Level

Glarsynost the alien lives on a platonic planet whose shape is that of a perfect regular dodecahedron.

Every day, she likes to go for a walk to have a look at her planet and see if she has any visitors.

What fraction of the planet's surface can she see from the middle of a face? From an edge? From a vertex?

Her walk needs to start and end at the same place, and she needs to be able to see every part of the planet's surface at some stage during her walk. Investigate the possible paths she could take. The challenge is to find the shortest path you can!

One way of investigating and recording this could be to create a net of a dodecahedron, and draw the path on the net, being careful to consider which faces will join when the net is folded up.

Here are two nets you could use, but you may find it easier to visualise an efficient path using a different one.

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?