Maths question from Laboratory Attendant exam, 2026 by JKSSB
A sphere is placed in a cube so that it touches all the faces of the cube. If 'a' is the ratio of the volume of the cube to the volume of the sphere and 'b' is the ratio of the surface area of the sphere to the surface area of the cube, then the value of 'ab' is:
Last updated Jun 13, 2026
Correct Answer:
Option D —
1
Step-by-Step Breakdown:
Relating the Dimensions:
Let the side length of the cube be x.
Since the sphere perfectly touches all faces of the cube, the diameter of the sphere is equal to the side length of the cube (x). Therefore, the radius of the sphere is half of the side length, which is x divided by 2.
Finding Ratio 'a' (Volume of Cube to Volume of Sphere):
The volume of a cube is calculated by cubing its side length: x cubed.
The volume of a sphere is four-thirds times pi times the radius cubed. Substituting the radius (x divided by 2), the radius cubed becomes x cubed divided by 8.
Multiplying four-thirds by x cubed divided by 8 simplifies the sphere's volume to: pi times x cubed divided by 6.
Now, we find ratio 'a' by dividing the cube's volume by the sphere's volume: x cubed divided by (pi times x cubed divided by 6). The x cubed terms cancel each other out, leaving 'a' equal to 6 divided by pi.
Finding Ratio 'b' (Surface Area of Sphere to Surface Area of Cube):
The surface area of a sphere is 4 times pi times the radius squared. Substituting the radius (x divided by 2), the radius squared becomes x squared divided by 4. Multiplying this by 4 times pi gives a sphere surface area of: pi times x squared.
The surface area of a cube is 6 times the area of one face: 6 times x squared.
Now, we find ratio 'b' by dividing the sphere's surface area by the cube's surface area: (pi times x squared) divided by (6 times x squared). The x squared terms cancel each other out, leaving 'b' equal to pi divided by 6.
Calculating the Value of 'ab':
To find 'ab', we multiply the two ratios together:
'ab' equals (6 divided by pi) times (pi divided by 6).
Notice that the 6 in the numerator cancels out the 6 in the denominator, and the pi in the numerator cancels out the pi in the denominator.
This leaves us with a final result of 1.
Answer verified by Quintessence Classes faculty — Karan Nagar, Srinagar.
