# Earth Curvature Calculator

Calculate Earth's curvature drop over any distance. Find how much of a distant object is hidden below the horizon based on observer height and distance.

## What this calculates

The Earth curves away from a flat line at roughly 8 inches per mile squared. Over 10 miles, that adds up to about 66 feet of drop. This calculator shows you the curvature drop at any distance, how far you can see to the horizon, and how much of a distant building or landmark is hidden below the curve.

## Inputs

- **Distance to Object** (mi) — min 0 — Distance from you to the object across the surface.
- **Observer Eye Height** (ft) — min 0 — Height of your eyes above the surface (or camera height).
- **Target Object Height** (ft) — min 0 — Total height of the distant object (e.g., building or tower).

## Outputs

- **Curvature Drop** (ft) — How far the surface curves away from a flat plane at that distance.
- **Curvature Drop** (m) — Curvature drop in meters.
- **Distance to Horizon** (mi) — How far you can see to the geometric horizon.
- **Hidden Height** (ft) — How much of the target is hidden below the horizon.
- **Visible Height** (ft) — How much of the target is still visible above the horizon.

## Details

**The curvature drop formula:**

h = d² / (2R)

Where h is the drop in feet, d is the distance in feet, and R is the Earth's mean radius (about 3,958.8 miles or 20,902,231 feet). This approximation works well for distances under a few hundred miles.

**Distance to horizon:**

d = √(2Rh)

Where h is your eye height above the surface. Standing at 6 feet tall, your horizon is about 3 miles away. From the top of a 100-foot lighthouse, you can see about 12.2 miles.

**Hidden height of distant objects:**

When an object is beyond your horizon, part of it disappears bottom-first. The hidden amount depends on how far beyond the horizon the object sits. For example, a 100-foot building at 20 miles is almost entirely hidden for a 6-foot observer since the horizon is only about 3 miles away and the curvature hides roughly 175 feet at that beyond-horizon distance.

**Important notes:**
- These are geometric calculations. Atmospheric refraction typically bends light downward, letting you see about 8% farther than the geometric horizon.
- Results assume a smooth surface (no terrain features).
- The formula uses Earth's mean radius and treats it as a perfect sphere, which is close enough for practical purposes.

## Frequently Asked Questions

**Q: How much does the Earth curve per mile?**

A: The Earth curves approximately 8 inches over the first mile. But the drop grows with the square of the distance: 2 miles gives about 2.67 feet, 5 miles gives about 16.6 feet, and 10 miles gives about 66.5 feet. The "8 inches per mile squared" rule is a handy approximation.

**Q: How far can you see on a clear day?**

A: Standing at 6 feet tall on a flat surface, the geometric horizon is about 3 miles away. From 100 feet up (like a ship's crow's nest), you can see about 12.2 miles. Atmospheric refraction typically extends this by about 8%, so real-world visibility is slightly farther.

**Q: Does refraction affect these calculations?**

A: Yes. Standard atmospheric refraction bends light around the curve slightly, effectively extending your line of sight by about 8%. Some calculators apply a refraction coefficient of about 7/6 to the Earth's radius to account for this. This calculator shows geometric values without refraction.

**Q: Can I use this for surveying or engineering?**

A: This calculator provides good estimates for educational and recreational use. Professional surveying accounts for local geoid shape, atmospheric conditions, and uses more precise models than a simple sphere. For distances under 10 miles, the spherical approximation is within a few inches of reality.

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Source: https://vastcalc.com/calculators/physics/earth-curvature
Category: Physics
Last updated: 2026-04-08