Which is the longest stick that can be contained inside a cube with side length 1 m? Please show calculation!

Asked on by Jackly

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tiburtius's profile pic

tiburtius | High School Teacher | (Level 2) Educator

Posted on

What you are searching is the longest line inside the cube which is of course space diagonal.

Formula for the length of space diagonal of a cuboid is


where `a,` `b` and `c` are lengths of the edges. Since cube has all equal sides formula for space diagonal of a cube is


Hence if `a=1"m"` then 

`d=1sqrt3=sqrt3approx1.732`  ` `

The length of the longest stick that can be containdes in cube with side length of 1m is `sqrt3"m."`

chrisyhsun's profile pic

chrisyhsun | Student, College Freshman | (Level 1) Salutatorian

Posted on

The longest stick that can be contained inside the cube will span the longest possible distance within the cube. This is the case when one end is at a given corner (let's call it A) and the other end is at the opposite corner. It may be a bit difficult to visualize what the "opposite corner" is on a cube, but if you position the cube so that vertex A is facing you head-on, the opposite will be the only corner you can't see from your vantage point.

Now, if you can "see" which two corners the stick spans, you'll be able to tell that this isn't just a problem of simple Pythagorean theorem. You actually have to essentially do the Pythagorean theorem twice - once "along" one face of the cube and another "along" a different face to get to the far corner. Since all of the edges are of length one, the calculation goes like this:

Pythagorean Theorem First Time:

1^2 + 1^2 = x^2

1 + 1 = x^2

2 = x^2 for now, we will keep it in this format, just to make the next calculation easier

Pythagorean Theorem Second Time: this time, though, instead of using two of the edges, we will use the answer from the previous calculation and another edge.

x^2 + 1^2 = y^2 (this means y is actually our final answer, not x!)

2 + 1 = y^2

3 = y^2

Taking the square root of both sides, we get that the length of the stick is square root of 3 meters.

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