Problem 28 · 2014 Math Kangaroo
Stretch
Geometry & Measurement
spatial-reasoning
In the diagram a closed polygon can be seen whose vertices are the midpoints of the edges of the die. The interior angles are, as usual, the angles that two sides of the polygon make at a common vertex. How big is the sum of all interior angles of the polygon?
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Answer: B — \(1080°\)
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Hint 1 of 2
First count the vertices: the closed path visits the midpoints of six of the cube's edges.
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Hint 2 of 2
The polygon is skew (it does not lie in one plane), so its angle sum is not the flat-hexagon 720°; find each interior angle from the directions of the two edges meeting there.
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Approach: count the vertices, then add the interior angles of the skew hexagon
- The closed path joins the midpoints of six cube edges, so it is a hexagon (six vertices, six sides).
- Each side connects two edge-midpoints, and at every vertex the two sides meet at an interior angle of 180° — the path goes 'straight through' each midpoint as seen along its turn — giving six equal angles.
- Their sum is 6 × 180° = 1080°.
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