Fresnel zone radius calculator: formula and worked example

Every path profile starts with the same question: how wide is the first Fresnel zone at the point where that tree line sits. Misjudge that radius and a crew ends up trimming a stand that was never close to the signal, while a stand that is close keeps growing until the link fades.

The formula

The first Fresnel zone radius at any point along a link is:

r = 17.3 × √(d1 × d2 / (f × D))

where r is in meters, d1 and d2 are the distances from each end to the point you're checking (in km), D is the total hop length (in km), and f is frequency in GHz. This is the standard metric form for the first Fresnel zone (n = 1), which is the one that matters for clearance.

A few things trip people up on this formula. d1 + d2 always equals D, so you're really just picking a fraction along the hop and plugging in the two legs. The radius changes along the path: it's zero at both towers and reaches its maximum at the midpoint, which is why the path profile diagram shows that classic bulge shape.

Worked example

Take an 11 GHz hop running 15 km point-to-point.

At the midpoint, d1 = d2 = 7.5 km, D = 15 km, f = 11 GHz:

r = 17.3 × √((7.5 × 7.5) / (11 × 15)) r = 17.3 × √(56.25 / 165) r = 17.3 × √0.341 r = 17.3 × 0.584 r ≈ 10.1 m

So at the halfway point on this hop, the full first Fresnel zone has a radius of roughly 10 meters. That's the number you'd compare against ground elevation, tower height, and anything growing in between.

Now check a point closer to one end, say 3.75 km from tower A (a quarter of the way along), so d1 = 3.75 km, d2 = 11.25 km:

r = 17.3 × √((3.75 × 11.25) / (11 × 15)) r = 17.3 × √(42.19 / 165) r = 17.3 × √0.256 r = 17.3 × 0.506 r ≈ 8.75 m

Smaller radius, as expected, since you're nearer the antenna. This is the step most back-of-envelope checks skip: run the formula at a few points along the hop, not just the midpoint, because a ridge or a stand of trees a third of the way in can matter more than what's sitting dead center.

Why 60% clearance, not 100%

Most link budgets don't require full first Fresnel zone clearance. The working rule of thumb is 60% of r1 at the obstruction point, which accounts for the fact that a fully blocked Fresnel zone causes measurable loss, but partial intrusion into the outer part of it mostly doesn't. For the 10.1 m midpoint result above, that puts the clearance target at about 6.1 m above the obstruction, not 10.1 m. Confirm the clearance criterion your link budget actually specifies before trimming to it. Some planners run 1.0, others run 0.6, and backhaul vendors don't always agree.

This is also where the formula runs out of usefulness on its own. It tells you the radius you need clear at a given point. It doesn't tell you how tall the canopy under that point is, or how fast it's closing the gap between now and next year's growing season. That's a field measurement, or a canopy-height read from an annual pass, not something you get from frequency and hop length alone.

If you're running these numbers hop by hop against guessed-at tree heights, Link Vegetation Monitor flags exactly where canopy height is intruding on your Fresnel clearance envelope, link by link, once a year, so the formula above tells you the threshold and the pass tells you which hops are already crossing it.