Angles of Elevation and Depression

Angles of elevation and depression are the most common way trigonometry shows up in real life. Whether you are estimating the height of a building, figuring out how far away a ship is, or calculating the angle for a safety ladder, the setup is always the same: a right triangle with a known angle measured from the horizontal.

This page builds directly on Finding Missing Sides and Finding Missing Angles — the difference is learning to recognize these triangles in the real world.

What Are Angles of Elevation and Depression?

Both angles are measured from the horizontal — this is the single most important thing to remember.

Angle of elevation: the angle between the horizontal line of sight and a line looking up to an object above you.

Angle of depression: the angle between the horizontal line of sight and a line looking down to an object below you.

A common mistake is measuring the angle from the vertical. If someone says “the angle of elevation is 35 degrees,” they mean 35 degrees above horizontal — not 35 degrees from straight up. This matters because it changes which trig ratio you use and completely changes the answer.

The Key Relationship: Elevation Equals Depression

When you look up at something, the angle of elevation from your position equals the angle of depression from that object back down to you. This follows from alternate interior angles formed by two horizontal lines (yours and the object’s) cut by the line of sight.

This is useful because problems often give you the angle of depression from one position, but the right triangle you actually solve uses that same angle as the angle of elevation from the other position.

Worked Example 1: Finding a Height (Angle of Elevation)

Problem: You are standing 50 feet from the base of a cell tower. The angle of elevation from your position to the top of the tower is 65 degrees. How tall is the tower?

Setup: The tower height is opposite the 65-degree angle. The 50-foot distance is adjacent. Use tangent.

$\tan(65°) = \frac{\text{height}}{50}$

$\text{height} = 50 \times \tan(65°) = 50 \times 2.1445 = 107.2 \text{ ft}$

Answer: The tower is approximately 107.2 feet tall.

Worked Example 2: Finding a Distance (Angle of Depression)

Problem: From the top of a 120-foot cliff, a rescue worker spots a boat at a 28-degree angle of depression. How far is the boat from the base of the cliff?

Setup: The angle of depression from the cliff top is 28 degrees. By alternate interior angles, the angle of elevation from the boat is also 28 degrees.

In the right triangle:

• The cliff height (120 ft) is opposite the 28-degree angle at the boat
• The horizontal distance $d$ is adjacent to that 28-degree angle

$\tan(28°) = \frac{120}{d}$

$d = \frac{120}{\tan(28°)} = \frac{120}{0.5317} = 225.7 \text{ ft}$

Answer: The boat is approximately 225.7 feet from the base of the cliff.

Common mistake here: Students often put the 28-degree angle inside the triangle at the cliff top, giving them $\tan(28°) = d/120$ and a wrong answer of 63.8 ft. The angle of depression is measured from the horizontal at the top, so you need to either use the alternate interior angle at the bottom, or recognize that the angle inside the triangle at the top is actually $90° - 28° = 62°$.

Worked Example 3: Two Observation Points

Problem: From two points on level ground that are 100 meters apart, the angles of elevation to a mountain peak are 35 degrees and 52 degrees. Find the height of the mountain.

Setup: Let $d$ = horizontal distance from Q (the closer point) to the base of the mountain. Then the distance from P to the base is $d + 100$.

From point Q: $\tan(52°) = \dfrac{h}{d}$, so $h = d \times \tan(52°)$

From point P: $\tan(35°) = \dfrac{h}{d + 100}$, so $h = (d + 100) \times \tan(35°)$

Set the two expressions for $h$ equal:

$d \times \tan(52°) = (d + 100) \times \tan(35°)$

$1.2799d = 0.7002d + 70.02$

$0.5797d = 70.02$

$d = 120.8 \text{ m}$

Now find the height:

$h = 120.8 \times \tan(52°) = 120.8 \times 1.2799 = 154.6 \text{ m}$

Verify: $h / (120.8 + 100) = 154.6 / 220.8 = 0.7002 = \tan(35°)$ ✓

Answer: The mountain is approximately 154.6 meters tall.

Real-World Application: Safe Tree Felling

A tree worker needs to determine whether a tree will hit a nearby building when felled. Standing 30 meters from the base of the tree, the worker (eye level 1.7 m) measures the angle of elevation to the top as 37.8 degrees. The building is 20 meters behind the worker (50 meters from the tree base).

First, calculate the height above eye level:

$h_{\text{above eyes}} = 30 \times \tan(37.8°) \approx 30 \times 0.7757 \approx 23.3 \text{ m}$

Then add the observer’s eye height:

$h_{\text{tree}} = 23.3 + 1.7 = 25.0 \text{ m}$

Since the tree is 25 meters tall and the building is 50 meters away, the tree will not reach the building — it would need to be at least 50 meters tall to pose a threat. Safe to proceed.

This is a quick field check that avoids the need for precise measurement tools. Any time you can measure the distance to an object and estimate the angle, you can calculate the height.

Common Mistakes

1. Measuring from the vertical instead of the horizontal. An angle of elevation of 35 degrees means 35 degrees above horizontal — not 35 degrees from straight up (which would be 55 degrees from horizontal). Mixing these up gives very wrong answers.
2. Placing the angle of depression inside the triangle wrong. When a problem gives a 28-degree angle of depression from the top, the angle inside the right triangle at the top is $90° - 28° = 62°$, not 28 degrees. Use alternate interior angles to place the 28 degrees at the bottom instead.
3. Forgetting the observer’s height. If a 5.5-foot-tall person measures the angle of elevation to a rooftop, the calculated height gives the distance from their eyes to the roof — add their height to get the true building height.
4. Using the wrong trig ratio. In most elevation/depression problems, the vertical and horizontal distances form the opposite and adjacent sides. That means you almost always use tangent — not sine or cosine. See When to Use Sine, Cosine, or Tangent for a refresher.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Angles of elevation and depression are both measured from the horizontal — never from the vertical
• The angle of elevation from point A to point B equals the angle of depression from B to A (alternate interior angles)
• Most elevation/depression problems use tangent because you are working with opposite (height) and adjacent (distance) sides
• For two-observation-point problems, set up two tangent equations with the same height $h$ and solve the system
• Always check whether the observer’s own height needs to be added to the calculated result

Return to Trigonometry for more topics in this section.