Skip to main content

LOS distance: 45 m and 25 m antennas

Short answer

Using d = √(2R ht) + √(2R hr) with R=6378 km, d = √(2×6.378×10⁶×45) + √(2×6.378×10⁶×25) ≈ 41.8 km.

Find out the line of sight distance between the transmitting antenna and receiving antenna if the transmitting antenna height is 45 m and the receiving antenna height is 25 m. [Given: the radius of Earth is 6,378 km].

The line-of-sight (LOS) distance is the sum of the distances from each antenna to the horizon. From the geometry of a tangent to a sphere of radius R:

   Tx                         Rx
   |\                         /|
 ht| \         d1   d2       / |hr
   |  `-.______  P  ______.-'  |
         earth (radius R)
 (R + h)² = R² + d²  →  d = √(2Rh + h²) ≈ √(2Rh)

Given: ht = 45 m, hr = 25 m, R = 6378 km = 6.378 × 10⁶ m.

d1 = √(2 R ht) = √(2 × 6.378×10⁶ × 45)
   = √(5.740×10⁸) = 23 959 m = 23.96 km
d2 = √(2 R hr) = √(2 × 6.378×10⁶ × 25)
   = √(3.189×10⁸) = 17 858 m = 17.86 km
d  = d1 + d2 = 23.96 + 17.86 = 41.82 km

Answer: LOS distance ≈ 41.8 km (using the given geometric earth radius).

Note: if standard atmospheric refraction is included (effective radius 4/3 × 6378 = 8504 km), d = 27.67 + 20.62 = 48.29 km.

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Propagation and Antenna chapter 4 questions →