Library / Geometry, coordinates, surveying and trigonometry / GEO-34
Geometry, coordinates, surveying and trigonometry
Height from two angles of elevation
Height of a tower or tree above level ground from two sightings on the same line, a distance d apart.
- Ref
- GEO-34
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- H [m]
Two sightings on a straight line towards a tower, a distance d apart, give the tower’s height above eye level. With elevations α from the farther point and β from the nearer, the horizontal distances satisfy x/tan α − x/tan β ... so the height above the eye is d tan α tan β/(tan β − tan α). Adding the eye height h gives the height above the ground.
Height of a tower or tree above level ground from two sightings on the same line, a distance d apart. α is the angle at the farther point and β the nearer, with α < β, and h is the eye height.
Calculator
- Height above the eye level, d tan α tan β / (tan β − tan α)8.66 m
- Horizontal distance from the nearer point to the tower5 m
- Horizontal distance from the farther point to the tower15 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With sightings of 30° and 60° from points 10 m apart and an eye height of 1.5 m, the top is 10.16 m above the ground, 5 m from the nearer point.
- If the two angles are close, the denominator is small and a degree of error in either angle changes the height a lot, so use a long baseline.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| α Elevation at the farther point | 5 to 59 ° | 2.421 | 426.8 | 424.4 | +10% | +17 % |
| β Elevation at the nearer point | 31 to 85 ° | 7.581 | 149.1 | 141.5 | +10% | −8.75 % |
| d Distance between the two observation points | 1 to 100 m | 2.366 | 88.1 | 85.74 | +10% | +8.52 % |
| h Eye height above the ground | 0 to 3 m | 8.66 | 11.66 | 3 | +10% | +1.48 % |
Sweep of Distance between the two observation points
| Distance between the two observation points (m) | Height above the ground (m) |
|---|---|
| 1 | 2.366 |
| 13.38 | 13.08 |
| 25.75 | 23.8 |
| 38.13 | 34.52 |
| 50.5 | 45.23 |
| 62.88 | 55.95 |
| 75.25 | 66.67 |
| 87.63 | 77.39 |
| 100 | 88.1 |
Sweep one variable
Variables and how each one contributes
Height above the ground
What the equation gives you. Everything else feeds it.
Eye height above the ground
The height of the observer’s eye (or the instrument) above the ground, in m. Raises the answer by the same amount. Raising it by 0.1 (to 1.6) moves the result by +0.984 %, from 10.16 to 10.26 m.
Distance between the two observation points
The distance between the two observation points, along the line to the tower, in m. A longer baseline gives a taller tower in direct proportion. Raising it by 10 % (to 11) moves the result by +8.52 %, from 10.16 to 11.03 m.
Elevation at the farther point
The angle of elevation of the top from the farther point, in degrees. A bigger angle at the far point means a nearer, lower top, so the height falls. Raising it by 1 (to 31) moves the result by +5.31 %, from 10.16 to 10.7 m.
Elevation at the nearer point
The angle of elevation of the top from the nearer point, in degrees. It must be bigger than the far angle. A bigger angle at the near point means a taller top for the same baseline. Raising it by 1 (to 61) moves the result by −1.67 %, from 10.16 to 9.991 m.
Worked example with the values above
- tan α = tan 30° = 0.57735 and tan β = tan 60° = 1.7321.
- Height above the eye: 10 × 0.57735 × 1.7321 / (1.7321 − 0.57735) = 8.6603 m.
- H = 1.5 + 8.6603 = 10.16 m.
Assumptions and limits
- A local flat plane with parallel north lines at all stations, and bearings measured clockwise from north.
- Distances are straight-line distances on level ground, in one consistent unit.
Common mistakes
- Measuring a bearing anticlockwise or from east instead of clockwise from north.
- Forgetting to take the result modulo 360° after adding or subtracting.
- Using the angle of depression or a flat-ground assumption when the ground is sloping.
Statistics and uncertainty
Angle errors dominate: a 1° error in a bearing moves a point 1.7 % of the distance sideways. Intersections of near-parallel sight lines amplify this, so keep the angle between the rays as large as practical.
Where it is used
- Navigation, orienteering and dead reckoning from legs and bearings.
- Surveying heights and positions with a compass, clinometer or theodolite.
Maths you need
- Bearing: angle clockwise from north, 000° to 360°
- Sine and cosine: east and north parts of a leg
- Right triangle trigonometry: sine, cosine and tangent