Friday, September 22, 2023

# Break Time…Cube inside a square!

Howdy, so this one I really enjoyed because it has so much practical applications…no? Ok, at least we should ALL know this! This question and solution comes from Quora. And I posted it also on my Quora blog, but don’t bother following me everywhere…this place is enough, so I’m posting it here as well.

Challenge: A cube rests inside a sphere so that each vertex touches the sphere. Can you find the volume of the cube, if the radius of the sphere is 6 cm? Solution: Although I had the solution myself, Orlando described his answer with perfect graphics and there’s no need to reinvent it! And proper attribution is due here.

So, let me walk you through it and using Orlando’s graphics.

In general, we know that volume of a Cube = side ^ 3 (where in this case, we’ll call the side “a”)

To solve the problem we have to find the value of a.  Then simply raise it to the power of 3 then voila! But to get to “a”, we need to take a few steps.

Let’s see, the diameter (D) = 2x radius. So, D=12cm…this is the diagonal of the cube.

Now comes the creative part! Take a look at the cube’s architecture (think in 3D)… Based on this figure, we have 2 triangles.  We now have 2 equations to solve for the triangles using good old Pythagorean theorems:

Pythagorean eq 1 (red triangle): c^2 = a^2 + a^2 = 2a^2

Pythagorean eq 2 (green triangle): D^2 = a^2 + c^2

12^2 = a^2 + 2a^2

144 = 3a^2

144/3 = a^2

a = SQRT(48)

a = 6.93cm

So, now we have the “holy grail”…the value of a.

So, can you compute the volume of the cube? 🙂