How to find the diagonal of a cube

Help Questions

Math › How to find the diagonal of a cube

Questions 1 - 2

1

Find the length of the diagonal of the following cube:

Length_of_diagonal

$4\sqrt{2}\ m$

$4\sqrt{3}\ m$

$4\ m$

$2\ m$

$8\ m$

Explanation

To find the length of the diagonal, use the formula for a triangle:

$a-a-a\sqrt{2}$

$4\ m-4\ m-4\sqrt{2}\ m$

The length of the diagonal is $4\sqrt{2}\ m$ .

2

Find the length of the diagonal connecting opposite corners of a cube with sides of length $2\sqrt{2}$ .

$2\sqrt6$ units

$6\sqrt{2}$ units

units

units

Explanation

Find the diagonal of one face of the cube using the Pythagorean Theorem applied to a triangle formed by two sides of that face ( and ) and the diagonal itself ():

$(2\sqrt{2})^2 + (2\sqrt{2})^2 = c^2$

$\rightarrow 8 + 8 = c^2$

$\rightarrow 16 = c^2$

$\rightarrow c = 4$

This diagonal is now the base of a new right triangle (call this ). The height of that triangle is an edge of the cube that runs perpendicular to this diagonal (call this ). The third side of the triangle formed by and is a line from one corner of the cube to the other, i.e., the cube's diagonal (call this ). Use the Pythagorean Theorem again with the triangle formed by , , and to find the length of this diagonal.

$(2\sqrt{2})^2 + (4)^2 = c^2$

$\rightarrow 8 + 16 = c^2$

$\rightarrow 24 = c^2$

$\rightarrow \sqrt{24} = c$

$\rightarrow c= 2\sqrt{6}$

Return to subject