Question

Body diagonal of a cube is 866 pm. Its edge length would be:
(A) 408 pm
(B) 1000 pm
(C) 500 pm
(D) 600 pm

Answer 1

Hint: In geometry a space diagonal of a polyhedron is a line connecting two vertices that are not on the same face.
Write the relationship between the body diagonal ‘d’ and the edge length ‘a’ of a cube:
\[{\text{d = }}\sqrt 3 \times {\text{a}}\]

Complete answer:
Consider the right angled triangle marked with blue colour,

Apply Pythagoras theorem and calculate the length of the hypotenuse.
\[{{\text{b}}^2}{\text{ = }}{{\text{a}}^2}{\text{ + }}{{\text{a}}^2}{\text{ = 2}}{{\text{a}}^2}{\text{ }} \\
{\text{b = }}\sqrt 2 {\text{a}} \\\]
Now consider the right angled triangle marked with blue colour
Apply Pythagoras theorem and calculate the length of the hypotenuse.
\[{{\text{d}}^2}{\text{ = }}{{\text{a}}^2}{\text{ + }}{{\text{b}}^2}{\text{ = }}{{\text{a}}^2}{\text{ + 2}}{{\text{a}}^2}{\text{ = 3}}{{\text{a}}^2}{\text{ }} \\
{\text{d = }}\sqrt 3 {\text{a}} \\\]
Write the relationship between the body diagonal ‘d’ and the edge length ‘a’ of a cube:
\[{\text{d = }}\sqrt 3 \times {\text{a}}\]
Rearrange the above equation to obtain the expression for the edge length:
\[{\text{a = }}\dfrac{{\text{d}}}{{\sqrt 3 }}\]… …(1)
Body diagonal of a cube is \[{\text{866 pm}}\] .
Substitute \[{\text{866 pm}}\] for the body diagonal ‘d’ in the equation (1).
\[{\text{a = }}\dfrac{{\text{d}}}{{\sqrt3 }} \\
{\text{a = }}\dfrac{{{\text{866 pm}}}}{{\sqrt 3 }} \\
{\text{a = }}500{\text{ pm}} \\\]
Thus, the edge length of the cube would be 500 pm.

Hence, the correct option is the option (C).

Note: During the derivation of the relationship between the body diagonal ‘d’ and the edge length ‘a’ of a cube, Pythagoras theorem is used. According to this theorem, for a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the remaining two sides.