Flat vs Round Earth Calculator - Horizon and Shadow Tests

A flat vs round earth calculator compares what three simple observations should show on a flat plane and on a spherical Earth: a second sunset from a higher viewpoint, a distant target hidden by curvature, and a circumference estimate from two stick shadows. It is useful for classroom demonstrations, science-fair planning, photography near the horizon, and checking whether a claimed observation uses the same geometry as the actual setup.

• • Second sunset checks: Estimate whether standing up, climbing stairs, or riding an elevator can recover a few seconds of a sunset after it disappears from a lower viewpoint.
• • Distant target visibility: Compare a target's height with the amount hidden below the curved-horizon tangent line at a known distance.
• • Stick-shadow circumference: Turn two noon shadow measurements into a central-angle difference, then scale a north-south baseline into a circumference estimate.
• • Lesson planning: Prepare input values that students can measure with a stopwatch, a meter stick, and a known map distance before discussing sources of error.

The calculator does not ask anyone to accept a conclusion from one number alone. Instead, it keeps the assumptions visible. A flat model has no curvature-hiding term, so the flat hidden-height output stays at zero. The spherical model uses radius, height, and angle relationships, so the hidden-height and sunset outputs change as the observer moves upward.

The most useful way to read the result is to compare patterns. If a higher eye point recovers part of the Sun, a farther target loses more of its bottom edge, and shadow angles scale with north-south distance, those three patterns all point to the same curved-surface geometry.

When you only need the target-hiding part of the comparison, the earth curvature calculator gives a narrower curvature workflow with drop-per-mile and obscured-height outputs.

This flat vs round earth calculator uses one spherical reference radius, optional effective-radius refraction, and three observation formulas. The sunset test converts extra horizon dip into seconds. The distant-target test computes the observer's horizon and the height hidden beyond it. The shadow test estimates a central angle from two stick-shadow ratios.

• R: effective Earth radius in meters or kilometers, using 6371 km for pure geometry and 7/6 of that value for the standard classroom refraction setting.
• h: observer height above the local horizon in the same length unit as R.
• d: surface distance from observer to target base, in kilometers.
• shadow angle: arctangent of shadow length divided by stick height, measured in degrees at each site.
• omega: Earth's angular rotation rate, used to convert a change in horizon dip into seconds of recovered sunset.

For the shadow experiment, the calculator turns each shadow into a solar zenith angle. A one-meter stick with no shadow at one site and a 0.1263 m shadow at another gives an angle difference of about 7.20 degrees. A north-south separation of 800 km then scales to roughly 40,009 km, close to the spherical reference circumference based on R = 6371 km.

All three outputs are approximations. They assume a smooth spherical Earth, a clear horizon, and measurements made along a north-south line for the shadow test. Local terrain and alignment errors can dominate a small classroom measurement.

According to NASA GSFC The Distance to the Horizon, distance to the horizon follows D = sqrt(2Rh) as a close approximation when observer height is much smaller than Earth's radius.

According to JPL Solar System Dynamics, Earth's mean radius is 6371.0084 km and its sidereal rotation period is 0.99726968 days.

For a focused horizon-distance setup with two-observer line of sight, use the distance to horizon calculator before bringing in sunset timing or stick shadows.

Four ideas explain why the outputs move the way they do. Each one connects a measurement you can make to a geometric quantity in the result panel.

A good experiment keeps these concepts separate. Use the sunset result to plan a timing test, the hidden-height result to choose a target tall enough to be seen, and the shadow result only when both sites are aligned close to north-south.

If the refraction setting is the part you want to isolate, the angle of refraction calculator shows how light bends between media before that bending is simplified into a horizon adjustment.

Use measured values where possible. The default numbers are classroom-friendly, but the flat vs round earth calculator is most useful when the input values come from the observation you plan to make.

1. 1 Set the sunset heights: Enter the lower eye height, the higher eye height, the time needed to move upward, and the observed sunset duration.
2. 2 Describe the distant target: Enter your eye height, the surface distance to the target, and the target's height above its base.
3. 3 Choose refraction: Use the standard setting for a rough real-air lesson, or choose none when you want the pure spherical geometry.
4. 4 Add shadow measurements: Enter the stick height, both shadow lengths, and the north-south separation between measurement sites.
5. 5 Compare outputs: Read the recovered sunset time, hidden height, flat hidden height, and circumference estimate as separate checks rather than one combined score.

For a beach demonstration, time a sunset from a seated eye height, stand quickly, and enter both heights. Then pick a low pier light or buoy at a known map distance and enter its height. The calculator will show whether the second sunset should be visible again and how much of the target should sit below the curved-horizon cutoff.

When the observation depends on the clock time of the Sun rather than only horizon geometry, the sunrise calculator gives coordinate-based solar timing for planning the session.

The calculator turns a debate-heavy topic into measurable assumptions and repeatable quantities. That makes it easier to plan, critique, and repeat a simple observation.

• • Keeps models separate: The result panel reports spherical hidden height and flat hidden height separately, so the comparison is visible instead of buried in prose.
• • Supports real measurements: Inputs match things a student can measure: height, time, shadow length, map distance, and target height.
• • Shows scale: Small height changes produce seconds of sunset recovery, while long target distances produce meters of hidden height.
• • Flags weak setups: A target inside the horizon or two equal shadow angles produce near-zero comparison value, which helps avoid wasted field work.
• • Encourages replication: Changing one input at a time makes it easier for a class or group to repeat the experiment with a different target, baseline, or viewpoint.

The main benefit is interpretability. A single photograph can be hard to judge because zoom, haze, tide, and elevation all matter. A calculation makes those assumptions explicit, then shows which measurement is driving the result.

Results from a flat vs round earth calculator depend on more than the ideal formulas. Treat the numbers as planning estimates, then record the conditions that could shift what you actually see.

• • The model treats Earth as a smooth sphere with radius 6371 km. It does not include the ellipsoid, local geoid height, terrain, waves, or camera lens distortion.
• • The refraction option is only a simplified effective-radius adjustment. Mirage layers, temperature inversions, and haze can shift or blur a horizon observation more than this setting suggests.

For a defensible field note, record the observer height, target height, distance source, time, weather, and whether the view crossed water or land. Those notes make it possible to separate geometry from local viewing conditions.

According to San Diego State University Atmospheric Refraction, standard refraction can make distant silhouetted objects visible beyond the simple geometric horizon.

For antenna and target-height work where radio propagation is the question, the radar horizon calculator applies the same line-of-sight idea to radar range.