Introduction
The Earth is round, so its surface bends away from you. That bend is called curvature. It is why a far-off ship looks like it sinks into the sea, and why the bottom of a distant building hides behind the water or land.
This Earth curvature calculator works out how much of that bend gets in your way. Type in your eye height, how far away the object is, and how tall the object is. The tool then tells you:
- Surface drop: how far the ground or water falls away over that distance
- Hidden height: how much of the object is blocked by the curve
- Visible height: how much of it you can still see
- Horizon distance: how far you can see before the Earth curves out of sight
Air bends light a little, which lets you see slightly farther than pure math says. This calculator shows both answers: the plain geometric one and the one adjusted for that bending, called refraction.
You can pick miles and feet or kilometers and meters, try quick setups like a beach, a mountain, or a plane, and compare two scenarios side by side. A moving diagram, a chart, and a step-by-step solution show you exactly how each number was found.
How to use our Earth Curvature Calculator
Enter your eye height, how far away your target is, and how tall that target is. The Earth curvature calculator then shows the surface drop, how much of the target is hidden by the curve, how much you can still see, and how far away your horizon sits.
Unit System: Pick Imperial (miles and feet) or Metric (kilometers and meters). All boxes and results switch to match.
Quick Scenarios: Click a preset like "Standing on a Beach" or "Commercial Airliner" to fill the boxes with ready-made values.
Observer Eye Height: Type how high your eyes or camera are above the ground or water.
Distance to Target: Type the flat ground distance between you and the object you are looking at.
Target Object Height: Type the full height of the far object, measured from its base to its top.
Advanced Mode: Turn this on to open extra boxes for air and camera settings.
Custom Planet Radius: Change Earth's radius if you want to test the Moon, Mars, or any other round body.
Air Temperature: Enter the air temperature where you stand. Warm and cold air bend light in different ways.
Atmospheric Pressure: Enter the air pressure at your spot. Higher pressure bends light a bit more.
Vertical Temperature Gradient: Enter how fast the air cools as you go up. A negative number means cooler air above you.
Camera Horizontal Field of View: Enter the side-to-side angle your lens or eye can see, in degrees.
Image Width (pixels): Enter how wide your photo or screen is in pixels, so the tool can show the horizon curve in pixels.
Compare Two Scenarios: Turn this on to add a second set of boxes, then fill in Scenario B's eye height, distance, and target height to compare both side by side.
Geometric + Refraction overlay: Keep this on to see the dashed line that shows how bending air changes the curve in the diagram.
Press Calculate to see your results, diagram, chart, and step-by-step math. Press Reset to start over with the default values.
What Is Earth Curvature?
Earth is a giant ball with a radius of about 3,959 miles (6,371 km). Because it is round, the ground or water between you and a faraway object slowly bends away from you. That bend is called Earth's curvature. It is why a ship far out at sea looks like it is sinking. The bottom of the hull slips behind the bulge of water while the mast still shows.
The Three Numbers That Matter
- Surface drop: how far the surface falls away over a set distance. Over 1 mile the drop is about 8 inches. Drop grows fast, because it follows a square rule, not a straight one.
- Horizon distance: the farthest spot on the surface you can see. Standing on a beach with your eyes 6 feet up, the horizon is only about 3 miles away. Climb higher and it moves out fast.
- Hidden height: how much of a far object is tucked behind the curve. Anything past your horizon has its base blocked, and only the top part stays in view.
Why Eye Height Changes Everything
Your own height above the ground has a big effect on the answer. At 6 feet the horizon sits about 3 miles out. At 100 feet it jumps to about 12 miles. From a plane at 35,000 feet it is roughly 230 miles. Raising your eyes lets you peek farther over the bulge, so less of a far building, mountain, or ship stays hidden.
How Air Bends Light
Air is not the same all the way up. It gets thinner and usually cooler with height, so light rays bend a little downward as they travel. This is called atmospheric refraction. It lets you see a bit past the true geometric horizon, normally about 8% farther. Warm air over cold water can bend light even more and make faraway things pop into view. That is why a simple geometry answer and a real-world view are never quite the same.
The Math Behind It
The math uses simple circle geometry. The distance to the horizon comes from a right triangle: d = √(2Rh + h²), where R is Earth's radius and h is your eye height. The surface drop over a distance uses the central angle: drop = R(1 − cos(d/R)). Refraction is handled with a trick: swap in a bigger "effective radius" so the curve looks flatter, which matches what your eyes really see.
Where People Use This
Sailors and pilots use horizon math for navigation and spotting other craft. Radio and cell engineers use it to plan tower height so signals clear the bulge. Surveyors correct long sight lines for curvature. Photographers use it to check when the horizon should look curved in a wide shot. At normal altitudes the bend is far too small to notice, which is why airplane window photos usually show a flat line.