What this calculator does
Find how tall something is, how far away it is, or the angle you look up or down to see it. The angle of elevation is how far you tilt your gaze up from level to see the top of something. The angle of depression is how far you tilt it down to see something below you. That angle, the distance along the ground and the height make a right triangle. So if you know any two, the tangent ratio finds the third. (The tangent is the tan key on a calculator; it compares the height to the ground distance.)
This angle of elevation calculator finds the height, the distance along the ground or the angle, looking up or down. It also allows for the height of your eye or measuring tool above the ground. And it handles the classic two-sighting problem. Say you can't reach the base of a tower or cliff. Measure the angle from two spots a known distance apart. The calculator finds the height and how far away the base is.
Where it's used
Why this math matters outside the classroom:
- Tree height. A forester 40 m from a tree reads a 35° angle with a clinometer and adds her 1.6 m eye height to get about 29.6 m.
- Lighthouse lookouts. A keeper 60 m above the sea sees a boat at a 12° angle of depression, so it is about 60 ÷ tan 12° ≈ 282 m away.
- Unreachable towers. Surveyors sight a mast at 30° and then at 45° after walking 20 m closer, finding its height of about 27.3 m.
- Aircraft approach. A plane descending on a 3° glide path is 1,000 ft up at about 1,000 ÷ tan 3° ≈ 19,081 ft from touchdown.
- Building codes. A planner checks that a 30 m tall building 45 m from a park sits at tan⁻¹(30 ÷ 45) ≈ 33.7° from the park edge.
Grade level and standards: Grades 9–10; Common Core State Standards for Mathematics HSG.SRT.C.8. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose what to find: the height, the horizontal distance, the angle, or the height from two sightings.
- Choose whether you're looking up (elevation) or down (depression).
- Enter the values you know. Measure distances along level ground. If you measured the angle standing up, enter your eye height.
- Read the answer, then the line-of-sight distance and the other measurements. The working shows the tangent step with your numbers.
What the inputs mean
- Horizontal distance
- Measured along level ground, not along the line of sight.
- Angle (degrees)
- Between the horizontal and the line of sight.
- Height
- Looking down, this is how far you are above the thing you're looking at. For example, a lighthouse lamp might be 60 m above the water.
- Eye (or instrument) height above the ground
- A clinometer (an angle-measuring tool) or phone app measures the angle from your eye, not from the ground. Adding your eye height gives the true height above the ground. For an adult standing, that's about 1.5 to 1.7 m.
- Angle from the nearer point, β (degrees)
- Walk toward the object in a straight line. The nearer angle β is always the larger one, because the top looks higher as you get closer.
Formula and method
tan θ = rise ÷ horizontal distance rise = distance × tan θ, distance = rise ÷ tan θ, θ = tan⁻¹(rise ÷ distance) height = rise + eye height line of sight = √(distance² + rise²) two sightings: rise = d × tan α × tan β ÷ (tan β − tan α)
Here's what the symbols mean. θ (theta) is the angle you measured. The rise is the height above your eye, and the horizontal distance is measured along level ground. tan⁻¹ is the inverse tangent, which turns a ratio back into an angle. In the two-sighting line, α (alpha) is the angle from the farther spot, β (beta) is the angle from the nearer spot, and d is the distance you walked between them.
The level ground and the upright line from the top meet at a right angle. Your line of sight is the hypotenuse, the longest side. The rise is the side across from the angle, and the ground distance is the side next to it. The tangent ratio links them: tan θ = opposite ÷ adjacent. Rearranging it gives each unknown in turn.
An angle of depression is measured down from your own level line of sight. That line runs parallel to the ground. So the angle of depression from the top equals the angle of elevation from the bottom back up. (They're called alternate angles.) That means the same triangle and the same formula work.
With two sightings, call the unknown rise h and the distance from the nearer spot x. From the nearer spot, x = h ÷ tan β. From the farther spot, x + d = h ÷ tan α. Subtracting gets rid of x and leaves d = h ÷ tan α − h ÷ tan β. Rearranging that gives the formula above.
Worked example
You can't reach the base of a radio mast across a fenced field. From one spot, the top is at a 30° angle of elevation. After walking 20 m straight toward it, the angle is 45°. How tall is the mast, measured from the ground (eye height 0)?
- Here α = 30°, β = 45° and d = 20, with tan 30° = 0.57735027 and tan 45° = 1.
- Height: h = 20 × 0.57735027 × 1 ÷ (1 − 0.57735027) = 27.3205 m.
- Distance from the nearer spot: x = 27.3205 ÷ tan 45° = 27.3205 m. So the farther spot was 47.3205 m away.
- Line of sight from the nearer spot: √(27.3205² + 27.3205²) = 38.637 m.
Result: The mast is about 27.3205 m tall, which is exactly 10(1 + √3) m. Its base is 27.3205 m from where you stopped.
Understanding the result
Small changes in a steep angle make big changes in height. Near 90° the tangent grows without limit. So measure from farther away, where the angle is between about 20° and 60°, for a more reliable answer.
The line-of-sight distance is always longer than the ground distance. Maybe you measured along a sloping line, for example with a laser rangefinder aimed at the top. That's the hypotenuse, not the ground distance. Use the right triangle calculator with the hypotenuse instead.
Assumptions and limitations
- The ground between you and the object is level, and the object stands straight up.
- In two-sighting mode, both spots and the base lie on one straight line, at the same ground level.
- Angles are in degrees, and all lengths use the same unit.
- The curve of the earth and the bending of light near the horizon are ignored. That only matters over several kilometers.
- Angles must be between 0° and 90°. For triangles without a right angle, use the triangle solver.
Frequently asked questions
What is the difference between an angle of elevation and an angle of depression?
Elevation is measured upward from level to something above you. Depression is measured downward from level to something below you. Say you're on a cliff top, looking down at a boat at 12°. Then the angle of elevation from the boat back up to you is also 12°.
How do I find the height of a tree with an angle of elevation?
Stand a known distance away on level ground and measure the angle to the top. Multiply the distance by the tangent of the angle, then add your eye height. At 40 m and 35°, the rise is 40 × tan 35° ≈ 28.01 m. With an eye height of 1.6 m, the tree is about 29.61 m tall.
How do I find the height when I can't measure the distance to the base?
Take two angle readings from spots a measured distance apart, on a straight line toward the object. The height is d × tan α × tan β ÷ (tan β − tan α). Here d is the distance between the spots, α is the farther (smaller) angle and β is the nearer one.
Does it matter which unit I use?
No, as long as every length uses the same one. Angles must be in degrees.
How this calculator is checked
Method: Right-triangle tangent ratio, alternate angles between parallel horizontal lines for the angle of depression, the Pythagorean theorem for the line of sight, and the standard two-observation height formula derived by eliminating the unknown distance. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
Embed this calculator
Put this Angle of Elevation Calculator on your own website or blog for free. Paste this code where you want it to appear: