Angle of Elevation and Height Calculator

How it works

The angle of elevation is the angle between the horizontal and your line of sight when you look up at something; looking down, the same angle below the horizontal is the angle of depression. The ground distance, the vertical rise and the line of sight form a right triangle with the right angle at the base of the object, so one side and one angle fix everything else by SOHCAHTOA.

Knowing the distance and the height, the tangent of the angle is rise over run, so the angle is the inverse tangent of (height − eye height) ÷ distance, and the line of sight is the hypotenuse √(distance² + rise²). Knowing the distance and the angle instead, the rise is distance × tan(angle), and the height of the object is that rise plus the height of your eye above the ground.

The distance and the heights can be in different units; everything is converted to metres internally and the results are reported back in the units you chose, so a tower a mile away can be measured in feet. The grade is the rise divided by the run as a percentage, which is how road and ramp slopes are quoted.

Formula

rise           = height − eye height
angle          = atan(rise ÷ distance)             (from distance and height)
rise           = distance × tan(angle)             (from distance and angle)
height         = eye height + rise
line of sight  = √(distance² + rise²)  =  distance ÷ cos(angle)
grade          = rise ÷ distance × 100%

Example

Standing 100 ft from a wall and looking at a point 50 ft up it (eye height 0), the angle of elevation is atan(50 ÷ 100) ≈ 26.5651° and the line of sight is √(100² + 50²) ≈ 111.8034 ft; the grade is 50%.

A tower 100 m away whose top is seen at an elevation of 60° from ground level rises 100 × tan 60° = 100√3 ≈ 173.2051 m, and the line of sight is 100 ÷ cos 60° = 200 m exactly.

Assumptions and limitations

• The ground between you and the object is level and the object is vertical, so the triangle has a true right angle at its base.
• The horizontal distance is measured along the ground to the point directly below the top, not the slanted line of sight.
• Over long distances the Earth's curvature lowers the apparent angle and atmospheric refraction raises it slightly, partly offsetting it; this calculator ignores both, which is fine for a building or a tree but not for a mountain 30 km away.
• A negative angle or a rise below eye level is reported as an angle of depression; the height can then come out below the observer's ground level.
• Units are converted with the exact definitions 1 ft = 0.3048 m and 1 mi = 1609.344 m; results are rounded for display only.

Frequently asked questions