Step 1: Recall the range of an antenna.
Because Earth is curved, a tall antenna of height $h$ can send signals only up to a line of sight distance: \[ d = \sqrt{2R_e h} \] where $R_e$ is Earth's radius.
Step 2: Write the service area.
The area covered is a circle of radius $d$: \[ A = \pi d^{2} = \pi(2R_e h) \]
Step 3: List the given values.
Antenna height $h = 81$ m and Earth radius $R_e = 6.4\times10^{6}$ m.
Step 4: Put values into the area.
\[ A = \pi\times2\times(6.4\times10^{6})\times81 \]
Step 5: Multiply the numbers.
\[ 2\times6.4 = 12.8, \qquad 12.8\times81 = 1036.8 \] \[ A = 3.1416\times1036.8\times10^{6} \approx 3257\times10^{6}\ \text{m}^{2} \]
Step 6: Convert to square kilometres.
Since $1\ \text{km}^{2} = 10^{6}\ \text{m}^{2}$: \[ A \approx 3257\ \text{km}^{2} \] \[ \boxed{A \approx 3257\ \text{km}^{2}} \]