On the Beach

Geometry Level pending

Three regular solids are sketched in the sand. If each square has a side 1 and accurately represents the size of the solid, what is the sum of their volumes?


The answer is 1.805.

Marta Reece by Marta Reece , 69, USA

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1 solution

Marta Reece
Mar 27, 2017

The first solid is a cube, with edge 1 1 and volume V c = 1 V_c=1 .

The second is on octahedron, seen point on. It too has an edge 1 1 , but the volume is V o = 2 3 V_o=\frac{\sqrt{2}}{3} .

The last is a tetrahedron, seed edge on. The edge is the single diagonal and it is a = 2 a=\sqrt{2} long. Its volume is V t = a 3 6 2 = 1 3 V_t=\frac{a^3}{6\sqrt{2}}=\frac{1}{3}

So the total volume is V = V c + V o + V t = 1 + 2 3 + 1 3 ≈ 1.805 V=V_c+V_o+V_t=1+\frac{\sqrt{2}}{3}+\frac{1}{3}\approx1.805 .

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Hm, I think it needs to be clarified what is actually happening.

For example, the apprentice could be seeing a (edit:) square based pyramid, from the base.

I originally thought that this was 3 different perspectives of the same solid.

Calvin Lin Staff - 4 years, 2 months ago

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I have rephrased the problem. I hope it helps.

Marta Reece - 4 years, 2 months ago

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Actually, I think I'm in error. I missed out the "regular solid". With that, I believe that these would be uniquely defined.

Calvin Lin Staff - 4 years, 2 months ago

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