leoxcvi wrote:
Excerpt from GMATClub Math Book:
Quote:
Cube inscribed in a sphere of radius r: Side of cube is \(\frac{2r}{\sqrt{3}}\)
My question is: How do you derive this logically? Step-by-step, without having to memorize the formula.
Also, how important is it to know these configurations (all 5 are on page 123 of the GMATClub Math Book)?
Cheers.
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2r is the diameter of the sphere, which is also the maximum aerial distance between two points within a cube.
a is the side of the cube.
a√2 is the physical distance on lower plane of the cube.
sides 2r, a√2 and a make a right angled triangle with 2r as the hypotenuse
\((a√2)^2 + a^2 = 4r^2\)
\(3a^2= 4r^2\)
\(a= \frac{2r}{\sqrt{3}}\)
It is not as important to memorize these formulas as to visualize how the figure fits. I think if you can visualize the figure most of these formulas will come from that figure without having to memorize any. As it says on the page these may appear in various forms on the GMAT, important to know how to apply these formulas to a question.