Why slant range, not depth
A transducer sends a pulse and times how long the echo takes to return. Sound travels through water at about 4,800 ft/s, so the unit computes range = speed × time / 2. That range is the straight-line distance to whatever reflected the pulse — the slant range. It is not the same as depth unless the fish is directly below you.
The formula is the Pythagorean theorem:
displayed = √(depth² + offset²)
At the nominal cone edge, where offset = depth × tan(cone/2), this collapses to:
edge display = depth / cos(cone/2)
On a 20° cone that is 1.5% over-reading at any depth. On a 40° cone it is 6.4%.
Edge error by cone angle and depth
How much deeper the sonar reads a fish at the nominal cone edge, in feet. The percentage version is the same for every depth — it is a function of the cone angle only.
| Cone | at 5 ft | at 10 ft | at 20 ft | at 30 ft | at 50 ft | at 100 ft | % over |
|---|---|---|---|---|---|---|---|
| 8° | <0.1 ft | <0.1 ft | <0.1 ft | <0.1 ft | 0.1 ft | 0.2 ft | 0.2% |
| 12° | <0.1 ft | <0.1 ft | 0.1 ft | 0.2 ft | 0.3 ft | 0.6 ft | 0.6% |
| 20° | <0.1 ft | 0.2 ft | 0.3 ft | 0.5 ft | 0.8 ft | 1.5 ft | 1.5% |
| 30° | 0.2 ft | 0.4 ft | 0.7 ft | 1.1 ft | 1.8 ft | 3.5 ft | 3.5% |
| 40° | 0.3 ft | 0.6 ft | 1.3 ft | 1.9 ft | 3.2 ft | 6.4 ft | 6.4% |
| 60° | 0.8 ft | 1.5 ft | 3.1 ft | 4.6 ft | 7.7 ft | 15.5 ft | 15.5% |
Error at the edge = depth × (1/cos(cone/2) − 1). For small angles the rule of thumb is roughly cone² ÷ 1650 percent.
The top of an arch is the most accurate reading
As the boat passes over a fish, the slant range starts large (fish at the leading edge of the cone, far away), shrinks to its minimum (fish directly below), then grows again as it exits the trailing edge. On the display that traces out an arch. The peak of the arch — the shortest range — is the reading closest to the fish's true depth, because the fish is most nearly below you at that moment.
The tails of the arch are the worst readings: the fish is at the largest angle then, so slant range is furthest from vertical depth. A fish 20 ft deep that passes 10 ft off the track reads as deep as 22.4 ft at its worst — an error of 2.4 ft. At the closest point (offset = 0) it would read exactly 20 ft.
What this means in practice
Three things follow directly from the geometry:
- Read the top, not the tails. The top of the arch is where the fish was closest to directly below you. If you want the depth to set your jig, read that number.
- Wide cones over-read more. A 60° cone has 15.5% error at the edge. A 12° cone has 0.6%. More coverage means more depth distortion at the edges — that is the tradeoff every wide-cone user accepts. Thecone-coverage calculator shows both widths.
- The fish was not moving sideways. A flat arch top does not mean the fish came straight under you — it means it was at its closest point during the pass. If it was off to the side that whole time, you never passed over it and the top-of-arch depth is still a slant read.
What this leaves out
- Speed of sound varies with temperature and salinity. The model uses 4,800 ft/s. In cold fresh water it is closer to 4,700; in warm salt water closer to 5,100. Most consumer fish finders use a fixed value, so there is a small systematic offset on top of the geometry.
- The transducer is not at the water surface. Transom mounts are 6–18 inches below the waterline. For depths over 10 ft the effect is under 1%, and this model ignores it.
- No beam pattern. This page shows pure slant-range geometry. Whether the fish paints at all at that angle depends on the cone and sensitivity; see thecoverage calculator.
More from the sonar model
- Why fish show as lines, not archesThe interactive simulator: drag a boat over a fish and watch the mark form.
- What speed should I drive for arches?Find the speed window where your depth and cone produce clean arches, not dashes or flat lines.
- Cone coverage calculatorHow wide your beam is at depth, and how much of it a given fish actually paints.
- Can you tell fish size from an arch?Why a bigger fish off to the side draws the same mark as a small one underneath.