Three circles each of radius 5 cm touch one another. The area (in cm²) subtended between them is:

The correct option is:
$25(\sqrt{3}-\frac{\pi }{2})$

Step-by-Step Derivation:

1. Identify the geometry of the centers of the circles:
Let the three mutually touching circles have centers A, B, and C. Each circle has a radius of $r=5\text{cm}$.
Since the circles touch each other externally, the distance between any two centers is equal to the sum of their radii:
Side length of triangle ABC = $r+r=2r=2\times 5=10\text{cm}$.
Therefore, the triangle formed by joining the centers of the circles is an equilateral triangle with a side length of $s=10\text{cm}$.

2. Calculate the area of the equilateral triangle:
The area of an equilateral triangle with side s is given by:
${\mathrm{Area}}_{\text{triangle}}=\frac{\sqrt{3}}{4}{s}^2$
Substituting $s=10\text{cm}$:
${\mathrm{Area}}_{\text{triangle}}=\frac{\sqrt{3}}{4}\times {10}^2=\frac{\sqrt{3}}{4}\times 100=25\sqrt{3}{\text{cm}}^2$

3. Calculate the area of the circular sectors inside the triangle:
Since the triangle is equilateral, each of its interior angles is ${60}^{\circ }$ (or $\frac{\pi }{3}$ radians).
The region inside the triangle occupied by the three circles consists of three sectors (one at each vertex of the triangle), each having a central angle of $\theta ={60}^{\circ }$ and a radius of $r=5\text{cm}$.
The sum of the areas of these three sectors is:
${\mathrm{Area}}_{\text{3 sectors}}=3\times \left(\frac{{60}^{\circ }}{{360}^{\circ }}\times \pi r^2\right)$
${\mathrm{Area}}_{\text{3 sectors}}=3\times \frac{1}{6}\times \pi \times 5^2=\frac{1}{2}\times 25\pi =\frac{25\pi }{2}{\text{cm}}^2$

4. Calculate the enclosed area subtended between the circles:
The area enclosed between the three circles is the area of the equilateral triangle minus the total area of the three sectors:
${\mathrm{Area}}_{\text{enclosed}}={\mathrm{Area}}_{\text{triangle}}-{\mathrm{Area}}_{\text{3 sectors}}$
${\mathrm{Area}}_{\text{enclosed}}=25\sqrt{3}-\frac{25\pi }{2}$
Factoring out 25:
${\mathrm{Area}}_{\text{enclosed}}=25\left(\sqrt{3}-\frac{\pi }{2}\right){\text{cm}}^2$