5 Pyramids in 1 Cube

Geometry Level 4

An artist is slicing an ice cube along the 3 diagonals joining 3 vertices, producing a piece of triangular ice pyramid as shown.

The artist then continues this process 3 more times on other diagonals, thereby creating 3 more pyramids, and finishes his artwork with a perfect tetrahedron in the end.

What is the fraction in volume of this tetrahedron compared to the original ice cube?

1 6 \frac16 1 5 \frac15 1 2 \frac12 1 3 \frac13 1 4 \frac14

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Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

The pyramid that the artist cuts out is a pyramid with the right triangle base, whose area is half of the cube's face. For a pyramid with square base (cube's face), the volume is 1 3 \dfrac{1}{3} of the cube's volume, so the volume of this right-triangular pyramid is half of one-third or 1 6 \dfrac{1}{6} of the cube's volume.

The artist proceeds 3 more times, so there are 4 such pyramids. Hence, the total volume = 4 × ( 1 / 6 ) 4\times(1/6) = 4 6 \dfrac{4}{6} = 2 3 \dfrac{2}{3} of the cube's volume.

As a result, the volume of the tetrahedron = 1 - 2 3 \dfrac{2}{3} = 1 3 \dfrac{1}{3} of the cube's volume.

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