Question
If the numerical value of the volume of a sphere is equal to the numerical value of its surface area, find the radius of the sphere.
This question was previously asked in
SSC CGL 2025 Tier 1 Question Paper (25-Sep-2025) (Shift 3)
The correct answer is
3 units
Sphere Calculations: Volume Equals Surface Area
This problem requires finding the radius of a sphere where its volume and surface area have the same numerical value. We will use the standard formulas for a sphere's volume and surface area.
Formulas for Sphere
- Volume ($V$): $V = \frac{4}{3}\pi r^3$
- Surface Area ($A$): $A = 4\pi r^2$
Here, '$r$' represents the radius of the sphere.
Equating Volume and Surface Area
The condition given is that the numerical value of the volume is equal to the numerical value of the surface area:
$V = A$
Substitute the formulas:
$\frac{4}{3}\pi r^3 = 4\pi r^2$
Solving for the Radius
To find the radius '$r$', we can simplify the equation. Assuming the radius is not zero ($r \neq 0$), we can divide both sides by common factors like $4\pi r^2$.
- Divide both sides by $4\pi$: $ \frac{1}{3} r^3 = r^2 $
- Divide both sides by $r^2$ (since $r \neq 0$): $ \frac{1}{3} r = 1 $
- Multiply by 3 to isolate '$r$': $ r = 3 $
Therefore, the radius of the sphere is 3 units.
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