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# Finding side length in a polyhedron, please explain

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Donatello starts with a marble cube of side length 10. He then slices a pyramid off each corner, so that in the resulting polyhedron, all the edges have the same side length s. Find s.

Jun 26, 2020

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Let the original side be     s+a+a=10

$$s^2=a^2+a^2\\ s^2=2a^2\\ a=\sqrt{\frac{s^2}{2}}$$

$$2\sqrt{\frac{s^2}{2}}+s=10\\ \sqrt{\frac{4s^2}{2}}+s=10\\ s*\sqrt{2}+s=10\\ s(\sqrt2 +1)=10\\ s=\frac{10}{1+\sqrt2}\\ s=\frac{10(1-\sqrt2)}{-1}\\ s=10(\sqrt2-1)$$

LaTex:

2\sqrt{\frac{s^2}{2}}+s=10\\
\sqrt{\frac{4s^2}{2}}+s=10\\
s*\sqrt{2}+s=10\\
s(\sqrt2 +1)=10\\
s=\frac{10}{1+\sqrt2}\\
s=\frac{10(1-\sqrt2)}{-1}\\
s=10(\sqrt2-1)

Jun 26, 2020
#3
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To find the side of this polyhedron, we may use an octagon.

s + 2( s /√2 ) = 10

2.414213s = 10

s = 4.142136

Jun 26, 2020