Cutting a Dodecahedron in Half

Geometry Level 2

A dodecahedron is shown in the image below resting on one its faces. If we were to pass a horizontal plane half-way between the lower base and upper base, the intersection will be a regular decagon (10-gon). Find the ratio of the side length of this decagon to the edge length of the dodecahedron.

cos ⁡ 5 4 ∘ \cos 54^{\circ} sin ⁡ 10 8 ∘ \sin 108^{\circ} sin ⁡ 5 4 ∘ \sin 54^{\circ} cos ⁡ 10 8 ∘ \cos 108^{\circ}

Hosam Hajjir by Hosam Hajjir , 56, Canada

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1 solution

Zico Quintina
May 6, 2018

A side of the decagon is the segment joining the midpoints of two adjacent edges of one of the side pentagonal faces of the dodecahedron. Referring to the above diagram, the ratio in question is d p = d 2 p 2 = sin ⁡ 5 4 ∘ \dfrac{d}{p} = \dfrac{\frac{d}{2}}{\frac{p}{2}} = \boxed{ \sin 54^{\circ}}

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