Horizon Distance Calculator: d = sqrt(2 R h)

How far away the horizon is, from the height of your eyes.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose which quantity you want to find.

Enter the other values in the units shown.

The calculator gives the answer with the formula, the substituted numbers, and the answer in other units.

Formulas used

  • d = \sqrt{2Rh}
  • h = \frac{d^2}{2R}

Worked example

With the default numbers in the calculator above:

  1. The formula, arranged to find the distance to the horizon: distance to the horizon = sqrt(2R) x height of the eye above the surface^0.5
  2. Put the values in (every value in its SI unit): distance to the horizon = sqrt(2R) x 1.7^0.5
  3. The answer: distance to the horizon: 4654.18 m
  4. The same answer in other units: 4.65418 km; 2.89197 miles; 2.51306 nautical miles

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: What do you want to find? Height of the eye above the surface; Distance to the horizon (leave it as it is if this is what you are finding) 4654.04 m.

  1. The formula, arranged to find the height of the eye above the surface: height of the eye above the surface = ((distance to the horizon) / (sqrt(2R)))^2
  2. Put the values in (every value in its SI unit): height of the eye above the surface = ((4654.04) / (sqrt(2R)))^2
  3. The answer: height of the eye above the surface: 1.6999 m
  4. The same answer in other units: 5.57709 feet

Common questions

How far is the horizon?

About 4.7 km for a person 1.7 m tall, and about 11 km from a height of 10 m.

How do I see a ship over the horizon?

Add the horizon distance for your eye height and the horizon distance for the height of the ship.

Why is the answer a little too short?

Air bends light slightly around the Earth. The horizon is about 6 percent farther than the simple formula says.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any geometry or navigation text (distance to the horizon).

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