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Height Distance Calculator

📅Last updated: December 2, 2025
✓Reviewed by: LumoCalculator Team

Calculate horizontal distance from height and angle of elevation using trigonometry. Also compute horizon distance based on observer height, or free fall time and impact velocity.

Height Distance Calculator

Trigonometry & distance calculations

Angle from horizontal (0° to 90°)

Calculation Results

Horizontal Distance
17.32 m
Height
10.00 m
Slope Distance
20.00 m
Angle
30.0° (0.5236 rad)
💡 Explanation

Using the angle of elevation (30.0°) and height (10.0m), the horizontal distance is calculated using tangent: distance = height / tan(angle). The direct line-of-sight (slope) distance is 20.0m.

Formula Used

distance = height / tan(angle)
distance = 10.00 / tan(30.0°) = 17.32 m

Unit Conversions

Height10.00 m = 32.81 ft
Distance17.32 m = 56.83 ft
Distance0.017 km = 0.011 mi

Related Calculators

Trigonometry Formulas

📐 Distance from Height
d = h / tan(θ)
  • • d = horizontal distance
  • • h = object height
  • • θ = angle of elevation
📏 Height from Distance
h = d × tan(θ)
  • • Measure distance to base
  • • Measure angle to top
  • • Add instrument height
🌍 Horizon Distance
d = √(2Rh)
  • • R = Earth radius (6,371 km)
  • • h = eye height above sea level
  • • Assumes spherical Earth
🎯 Free Fall
t = √(2h/g)
  • • g = 9.81 m/s² (gravity)
  • • v = √(2gh) impact velocity
  • • Ignores air resistance

Horizon Distance Reference

Observer HeightHorizon DistanceIn Miles
1.7 m (eye level)4.7 km2.9 mi
10 m (3rd floor)11.3 km7.0 mi
30 m (10th floor)19.6 km12.2 mi
100 m (skyscraper)35.7 km22.2 mi
1,000 m (mountain)113 km70 mi
10,000 m (airplane)357 km222 mi

* Geometric horizon only. Actual visibility depends on atmospheric conditions. Refraction can extend visible distance by ~8%.

Common Height References

Standing person1.7 m (5.6 ft)
Single-story building3 m (10 ft)
Telephone pole12 m (40 ft)
Lighthouse30 m (100 ft)
Statue of Liberty93 m (305 ft)
Big Ben96 m (316 ft)
Airplane cruising10000 m (33000 ft)

Practical Applications

🏗️ Surveying
  • • Building height measurement
  • • Land elevation mapping
  • • Construction planning
🚢 Navigation
  • • Lighthouse visibility range
  • • Ship detection distance
  • • Radar horizon calculation
📡 Telecommunications
  • • Antenna tower placement
  • • Line-of-sight coverage
  • • Signal propagation
🎓 Education
  • • Trigonometry problems
  • • Physics experiments
  • • Geometry demonstrations

Right Triangle Relationships

Trigonometric Ratios

sin(θ) = opposite / hypotenuse = h / s
cos(θ) = adjacent / hypotenuse = d / s
tan(θ) = opposite / adjacent = h / d

Pythagorean Theorem

s² = h² + d²
Where:
• h = height (opposite)
• d = distance (adjacent)
• s = slope distance (hypotenuse)

Frequently Asked Questions

How do I calculate distance from height and angle?
Use the tangent function: Distance = Height / tan(angle). For example, if an object is 10 meters tall and you measure an angle of elevation of 30°, the horizontal distance is 10 / tan(30°) = 10 / 0.577 ≈ 17.3 meters. This is basic trigonometry using a right triangle.
How far can I see to the horizon?
The horizon distance depends on your eye height above sea level. The formula is d = √(2Rh), where R is Earth's radius (6,371 km) and h is your height. At eye level (1.7m), the horizon is about 4.7 km away. From a 10-story building (30m), you can see about 19.6 km. From an airplane at 10,000m, the horizon is about 357 km away.
What is the slope distance vs horizontal distance?
Horizontal distance is the flat, ground-level distance between two points. Slope distance (or slant distance) is the actual straight-line distance through the air, which is always longer when there's elevation involved. They're related by: Slope Distance = Horizontal Distance / cos(angle) or Slope Distance = Height / sin(angle).
How long does an object take to fall from a height?
In free fall (ignoring air resistance), the time to fall is t = √(2h/g), where h is height and g is gravity (9.81 m/s²). For example, from 10 meters: t = √(2×10/9.81) ≈ 1.43 seconds. The impact velocity would be v = √(2gh) = √(2×9.81×10) ≈ 14 m/s (50 km/h).
How do surveyors measure building heights?
Surveyors use a theodolite or clinometer to measure the angle of elevation to the top of a building from a known distance. Using trigonometry (height = distance × tan(angle)), they calculate the height. For accuracy, they add the height of the measuring instrument. This is called indirect height measurement.
Why does the horizon appear further away on clear days?
The geometric horizon distance is fixed based on your height, but what you can actually see depends on visibility conditions. Atmospheric refraction can bend light, making the horizon appear about 8% further away than the geometric calculation. On very clear days with low humidity, you can see further than on hazy days.