Crystal Clear

Geometry Level 2

The jewelry shop sells its products cut from the same cubic block in 3 different shapes: a triangular-based prism, a square-based pyramid, and an octahedron. The prism has the base area equal to half the area of the cube's base while the octahedron has its 6 vertices on the centers of the cube's faces, and all 3 shapes have the same height as shown in the picture.

Which of these choices would give you more crystal?

I. A full triangular prism or
II. A pyramid plus an octahedron

Option I Option II Doesn't matter. Both options have got the same amount. Not enough information

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Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

Let V V be the volume of the original cube. Then it's obvious that the prism's volume = V 2 \dfrac{V}{2} because its base is half of the square.

Then the pyramid's volume = V 3 \dfrac{V}{3} for typical pyramid's formula.

Finally, for the octahedron, when we cut this shape in half, we will obtain 2 identical square-based pyramids, and each pyramid has half of the cube's height and half of the cube's base, for its diagonal equals to the cube's side length S S . Hence, base area = ( 1 2 ) ( S 2 ) (\frac{1}{2})(S^2) = half of the original square.

As a result, the volume of the octahedron = 2 × ( 1 2 ) × ( 1 2 ) × ( V 3 ) 2\times(\dfrac{1}{2})\times(\dfrac{1}{2})\times(\dfrac{V}{3}) = V 6 \dfrac{V}{6} .

Therefore, V 2 \dfrac{V}{2} = V 3 \dfrac{V}{3} + V 6 \dfrac{V}{6} , or both options have got the same amount of crystals.

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