Angle of Depression Calculator

What the angle of depression measures

The angle of depression is the angle between a horizontal line from your eye and your line of sight to something lower. OpenStax uses that wording in Contemporary Mathematics, section 10.8, published in 2023, and geometry classes use the same definition.

Two details do the work. The reference line is horizontal, not vertical, and the object sits below it. That puts the angle between 0 and 90 degrees, measured downward from level.

A lifeguard in a chair, a drone pilot over a roof, a surveyor on a riverbank and a passenger at a bridge railing all read this angle without naming it. It answers one question: how far below level do I look to see that thing? This page solves that downward angle. The angle of elevation page solves the same triangle from the ground up.

The three numbers in every problem

Every problem has the same three lengths. The height difference between eye and object is the opposite side. The level distance is the adjacent side. The line of sight is the hypotenuse.

Tangent does the main job: tan of the angle equals height divided by level distance. Sine and cosine cover the cases where the line of sight is the number you know. The angle comes from the inverse tangent, written atan or tan to the minus one.

Keep both distances in the same unit, and the length units cancel, so the angle comes out in degrees or radians. One degree is pi divided by 180 radians, or 0.017453, a definition NIST prints in its SI conversion guide. Spreadsheets work in radians, while a phone level shows degrees.

Example: a 60 foot cliff and a boat 90 feet out

A cliff rises 60 ft above the water and a boat sits 90 ft from the base. From the top, the angle of depression to the boat is the inverse tangent of 60 over 90.

60 divided by 90 is 0.6667, and atan of that is 33.69 degrees. The line of sight is the hypotenuse: 60 squared is 3,600, 90 squared is 8,100, and the square root of 11,700 is 108.17 ft.

Check the answer from the other end. Tan of 33.69 degrees is 0.6667, and 0.6667 times 90 is 60.0. If the boat drifts to 150 ft, the angle falls to 21.8 degrees. If it comes in to 30 ft, the angle climbs to 63.43 degrees, steep enough that the boat is almost under the cliff.

That last row is the grade of the line of sight. A 66.67 percent grade and a 33.69 degree angle describe the same slope, which is why the two units show up side by side on drawings.

Example: an airplane on a 3 degree path

Airplanes use the same angle on approach. The FAA's Aeronautical Information Manual says a visual glide path is normally set at 3 degrees, and an instrument glide slope is adjusted to the same angle. A pilot 400 ft above the runway threshold on that path is looking down 3 degrees.

Solve for the level distance: 400 divided by tan of 3 degrees gives 7,632.45 ft, or 1.45 miles. The line of sight is 7,642.93 ft, only 10 ft longer. At small angles the slope and the level distance are nearly the same number.

A shortcut for 3 degrees: the path drops 276.7 ft per mile, since tan of 3 degrees times 5,280 is 276.7. At 1,000 ft above the runway the aircraft is 19,081 ft out, or 3.61 miles.

The same numbers work at ground level. A wheelchair ramp at 4.76 degrees, a roof at 18.43 degrees and an aircraft at 3 degrees are all the same tangent, scaled to different lengths.

Working backward for height or distance

The same three formulas run in reverse, and that is where most real jobs sit. A roof inspector looking 12 degrees below level from 500 ft away wants the height. Height equals 500 times tan of 12 degrees, which is 106.28 ft, with a line of sight of 511.17 ft.

At 30 degrees and 200 ft of level ground, the height is 115.47 ft and the line of sight is 230.94 ft. At the same 200 ft, 12 degrees would give only 42.51 ft, so the steeper angle nearly triples the height.

If the line of sight is the number you have, multiply by sine to get the height and by cosine to get the level distance. A 108.17 ft sight at 33.69 degrees gives 60 ft and 90 ft, the cliff numbers from earlier.

Below about 5 degrees, the line of sight and the level distance differ by less than 0.4 percent, so either one works for a rough estimate. At 20 degrees the gap is 6.4 percent, and at 45 degrees it is 41 percent. That is when using the wrong one ruins the answer.

Depression and elevation are the same angle

Stand in the boat and look up at the cliff top. That angle of elevation is also 33.69 degrees. The horizontal line at the top and the horizontal line at the boat are parallel, and the line of sight cuts across both, so the two angles match. Geometry calls this the alternate interior angle rule.

The practical result: any elevation angle can be used as a depression angle, as long as you keep track of which end you stand on. Textbooks mix the two, and the arithmetic does not change.

Two other angles get confused with it. The angle from vertical is the complement, 90 minus 33.69, or 56.31 degrees. A bearing or azimuth is a compass direction and says nothing about how far down you look. The reference angle from trigonometry is measured from the x-axis, which is the same idea without an observer.

The angle of elevation calculator handles the looking up case with the same triangle, and its page covers ramps and ladders. The right triangle calculator and the trigonometry calculator solve the general triangle when the angle is not the unknown.

Degrees, percent grade and slope ratios

Percent grade is 100 times the tangent of the angle, so 33.69 degrees is a 66.67 percent grade. That sounds impossible for a road, and it is: 66.67 percent climbs 66.67 ft for every 100 ft forward, which is a ski run, not a highway.

The ratio form is friendlier. A 1:12 slope means 1 ft of rise per 12 ft of run, which is 8.33 percent and 4.76 degrees. That is the maximum running slope for a wheelchair ramp under the ADA Standards, section 405.2. A 4:12 roof pitch is 18.43 degrees, a 6:12 is 26.57, and a 12:12 is 45.

Use the run, not the line of sight, in that division. The cliff has a 66.67 percent grade over 90 ft of level water. Divide 60 by the 108.17 ft line of sight instead and the grade falls to 55.47 percent, and the angle that goes with it is 29.02 degrees, which is wrong.

Mistakes that bend the answer

Five mistakes account for most of the bad answers this page sees.

Measuring from vertical instead of horizontal is the first. At small angles the error is huge: an object 3 degrees below level is 87 degrees from vertical, and mixing the two turns a gentle look into a near vertical one.

Using the line of sight where the level distance belongs is the second. In the cliff example that turns 33.69 degrees into 29.02 degrees, and the distance you report comes out about 20 percent long.

Forgetting the observer's height is the third. A person 5.5 ft tall on that cliff has eyes at 65.5 ft, not 60, and the angle becomes 36.05 degrees. Surveyors add the instrument height for the same reason.

Ignoring the curve of the Earth matters over water. The drop is about 8 inches per mile squared, so 1 mile adds 8 inches and 10 miles add 66.7 ft, more than the height of this cliff. Below a mile it disappears into the rounding.

Last, check the calculator mode. A phone in radian mode takes the tangent of 33.69 radians, which is a different angle entirely. The numbers on this page are in degrees unless you ask for radians.