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A solid has a square base of side $s$ . The upper edge is parallel to the base and has length $2s$. All other edges have length $s$ . What is the volume of the solid ?

NB : The volume of the tetrahedron with all sides length l is $ V = \dfrac{\sqrt2}{12}l^3$

Way to infinity
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  • Obviously all the edges are straight lines... right? – pppqqq Jan 23 '14 at 08:47
  • Yes . You are right . – Way to infinity Jan 23 '14 at 09:05
  • is that a section of Square pyramid – Suraj M S Jan 23 '14 at 09:15
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    Especially given your "NB", it seems like you're describing a solid formed by slicing a regular tetrahedron by a plane parallel to (and half-way between) two opposite edges. In that case, the volume of the solid is half the volume of a regular tetrahedron of side $2s$. – Blue Jan 23 '14 at 09:31
  • Can you explain it ? I have been asked this question by my younger sister . – Way to infinity Jan 23 '14 at 09:34
  • @Waytoinfinity: Imagine a regular tetrahedron balancing on an edge, with the opposite edge parallel to the table. A plane parallel to the table will be parallel to those opposite edges. If the plane passes halfway between the table and the "upper" edge, then that plane will pass through the midpoints of the four remaining edges; those midpoints determine a square whose edges have half the length of the tetrahedron's edges (since each edge of the square is a "mid-line" of one of the tetrahedron's faces). And clearly the plane cuts the volume of the tetrahedron in half. – Blue Jan 23 '14 at 09:51
  • it will be clear to me if you provide this with necessary figure . – Way to infinity Jan 23 '14 at 09:52
  • @Waytoinfinity: Here's a cool animation: http://www.mmmlib.com/anne%20tyng%20maryland.html . If you search for "cut a tetrahedron in half", you'll find other images conveying the same idea. – Blue Jan 23 '14 at 09:54
  • So can you put your comments as an answer ? – Way to infinity Jan 23 '14 at 09:58

2 Answers2

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Converting comments to answer.

Especially given your "NB", it seems like you're describing a solid formed by slicing a regular tetrahedron by a plane parallel to (and half-way between) two opposite edges. In that case, the volume of the solid is half the volume of a regular tetrahedron of side $2s$.

Imagine a regular tetrahedron balancing on an edge, with the opposite edge parallel to the table. A plane parallel to the table will be parallel to those opposite edges. If the plane passes halfway between the table and the "upper" edge, then that plane will pass through the midpoints of the four remaining edges; those midpoints determine a square whose edges have half the length of the tetrahedron's edges (since each edge of the square is a "mid-line" of one of the tetrahedron's faces). And clearly the plane cuts the volume of the tetrahedron in half.

Here's a cool animation (though without the "balancing on an edge" thing):

enter image description here

Image credit: http://www.mmmlib.com/anne%20tyng%20maryland.html .

Blue
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We can divide the solid in two tetrahedra and a square pyramid. See fig.1.

TwoTetraAndPyr Fig.1

The pyramid height $GH$ has measure $x$, and the measure of $DB$ is the double of $x$. (Check it out).

Using pythagoras in $\triangle ABD$ we get: $$x=\frac{s \sqrt{2}}{2}$$ We know that the volume of the solid $V$ is: $$V=2V_{tetra}+V_{pyr}$$ where $V_{tetra}$ is the volume of each tetrahedron and $V_{pyr}$ is the volume of the square pyramid. Calculating each term we have: $$V=2 \frac{\sqrt{2}}{12}s^3+\frac{1}{3}s^2 \frac{s \sqrt{2}}{2} \Rightarrow$$ $$V= \frac{\sqrt{2}}{3}s^3$$

RicardoCruz
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