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

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 = }}\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.