I'm not sure that we do enough work on 3D geometry

a good end of term task - if not used elsewhere

looking at a mixture of (2D representations of) some solids

powerpoint

and some complex solids

powerpoint: potatoes

it is relatively easy to prove that Euler's rule is true for any prism

and also any anti-prism

powerpoint: prisms (and anti-prisms)

it is also relatively simple to prove that Euler's relationship is true for any pyramid

and dipyramid

powerpoint: pyramids and dipyramids

although Euler wasn't the first to develop the rule that has his name, he did attempt an interesting proof, by induction: the relationship works for a resulting basic shape (the tetrahedron) following a collapse and at all stages of all options when progressively 'collapsing' a solid

this proof was later shown to be flawed but is still interesting

powerpoint: Euler's rule

can be used to establish a rule for the sum of the interior angles of a polygon

the idea behind Euler's attempted proof by induction

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