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.
$$\begin{gathered} {\text{b}}^2\text{ = }{\text{a}}^2\text{ + }{\text{a}}^2\text{ = 2}{\text{a}}^2 \\ \text{b = }\sqrt{2}\text{a} \end{gathered}$$
Now consider the right angled triangle marked with blue colour
Apply Pythagoras theorem and calculate the length of the hypotenuse.
$$\begin{gathered} {\text{d}}^2\text{ = }{\text{a}}^2\text{ + }{\text{b}}^2\text{ = }{\text{a}}^2\text{ + 2}{\text{a}}^2\text{ = 3}{\text{a}}^2 \\ \text{d = }\sqrt{3}\text{a} \end{gathered}$$
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:
$$\text{a = }{\displaystyle \frac{\text{d}}{\sqrt{3}}}$$… …(1)
Body diagonal of a cube is $$\text{866 pm}$$ .
Substitute $$\text{866 pm}$$ for the body diagonal ‘d’ in the equation (1).
$$\begin{gathered} \text{a = }{\displaystyle \frac{\text{d}}{\sqrt{3}}} \\ \text{a = }{\displaystyle \frac{\text{866 pm}}{\sqrt{3}}} \\ \text{a = }500\text{ pm} \end{gathered}$$
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.