Right-Triangle Problem Types: Angles of Elevation, Depression, and Sloped Distances

What These Problems Have in Common

Angles of elevation, angles of depression, ladders, ramps, and sight lines all reduce to the same workflow: translate a real situation into a right triangle, then use the trig ratio that matches the sides you know and the side you need.

• Angle of elevation: measured upward from a horizontal line of sight (you look up).
• Angle of depression: measured downward from a horizontal line of sight (you look down).
• Sloped distance: a distance along a ramp, ladder, cable, or line of sight (often the hypotenuse of the right triangle).

Key diagramming habit: always mark the horizontal

In elevation/depression problems, the angle is always measured from a horizontal line. When you sketch, draw a small horizontal segment at the observer’s eye (or at the top point) to show what the angle is measured from. Then drop a vertical to form the right triangle.

Diagramming Checklist (Use Every Time)

1. Sketch the situation (a wall, ground, ramp, observer, object).
2. Draw the right triangle: ground is horizontal, height is vertical, slope/line-of-sight is diagonal.
3. Place the given angle at the observer (for elevation) or at the top point (for depression) measured from a horizontal line.
4. Label known lengths with units (m, ft, etc.).
5. Decide what each side represents: ground distance (horizontal), height (vertical), sloped distance (diagonal).
6. Choose a trig ratio that uses the known side(s) and the unknown side.
7. Solve and round appropriately, then state the answer with correct units.

Rounding and units guidance

• Unless told otherwise, round lengths to a sensible precision such as nearest tenth (0.1) or nearest whole unit, depending on the context.
• Keep units consistent. If the ground distance is in meters, the height you compute is in meters.
• Write your final answer with units: “12.4 m”, not just “12.4”.

Problem Type A: Find a Height (Angle of Elevation)

Example 1: Height of a building from a measured distance

You stand 50.0 m from the base of a building on level ground. The angle of elevation to the top is 28.0°. Find the building’s height.

Step-by-step

1. Draw the triangle: horizontal ground from you to the building is one leg; building height is the vertical leg; line of sight is the diagonal.
2. Mark the angle at your position between the horizontal ground line and the line of sight: 28.0°.
3. Label sides: adjacent (ground) = 50.0 m, opposite (height) = h.
4. Choose the ratio that connects opposite and adjacent: tan(θ) = opposite/adjacent.
5. Set up and solve:

tan(28.0°) = h / 50.0 m  h = 50.0 m · tan(28.0°)

Compute:

h ≈ 50.0 · 0.5317 ≈ 26.6 m

Answer: The building is approximately 26.6 m tall (to the nearest tenth of a meter).

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Common diagram mistake to avoid

Do not place the 28.0° at the building unless the problem states the angle is measured there. For angle of elevation, the angle is at the observer, measured from the observer’s horizontal.

Problem Type B: Find a Horizontal Distance (Angle of Elevation)

Example 2: How far from a tower?

A tower casts no helpful shadow (cloudy day), so you use a clinometer. The angle of elevation to the top is 35.0°. The tower’s height is known to be 40.0 m. How far are you from the base of the tower (horizontal distance)?

Step-by-step

1. Draw the triangle: height is vertical (40.0 m), ground distance is horizontal (d), line of sight is diagonal.
2. Angle location: at the observer on the ground, between horizontal and line of sight: 35.0°.
3. Identify sides relative to the angle: opposite = 40.0 m, adjacent = d.
4. Choose the ratio: tan(θ) = opposite/adjacent.
5. Set up and solve:

tan(35.0°) = 40.0 m / d  d = 40.0 m / tan(35.0°)

Compute:

d ≈ 40.0 / 0.7002 ≈ 57.1 m

Answer: You are approximately 57.1 m from the base of the tower.

Quick reasonableness check

A 35° angle is moderate; the distance should be somewhat larger than the height. 57.1 m is larger than 40.0 m, so it’s plausible.

Problem Type C: Find a Line-of-Sight (or Sloped) Distance

Example 3: Distance along a ramp

A wheelchair ramp rises 0.90 m vertically to a doorway. The ramp makes an angle of 6.0° with the ground. Find the length of the ramp surface (sloped distance).

Step-by-step

1. Draw the triangle: vertical rise = 0.90 m, ramp surface = L (diagonal), ground run is horizontal (unknown but not needed).
2. Angle location: at the bottom where the ramp meets the ground: 6.0°.
3. Identify sides: opposite = 0.90 m, hypotenuse = L.
4. Choose the ratio connecting opposite and hypotenuse: sin(θ) = opposite/hypotenuse.
5. Set up and solve:

sin(6.0°) = 0.90 m / L  L = 0.90 m / sin(6.0°)

Compute:

L ≈ 0.90 / 0.1045 ≈ 8.61 m

Answer: The ramp length is approximately 8.61 m (about 8.6 m to the nearest tenth).

Interpretation note

Small angles produce long ramps for a given rise; a 6° ramp being several meters long is expected.

Angles of Depression: Same Triangle, Different Viewpoint

An angle of depression is measured downward from a horizontal line at the observer’s eye (often at a higher point). The key fact for diagramming is that the observer’s horizontal line is parallel to the ground, so the angle of depression matches the corresponding angle of elevation in the right triangle you draw (after you create the vertical drop to the ground).

Example 4: Looking down from a cliff (find height)

A lookout point on a cliff observes a boat. The angle of depression to the boat is 12.0°. The horizontal distance from the base of the cliff to the boat is 200 m. Find the height of the cliff above the water (ignore observer’s eye height).

Step-by-step

1. Sketch: draw the cliff as a vertical line, water as horizontal. From the lookout, draw a horizontal reference line, then a line of sight down to the boat.
2. Form the right triangle: vertical leg = cliff height h, horizontal leg = 200 m.
3. Use the angle: the angle between the horizontal at the lookout and the line of sight is 12.0°. In the right-triangle setup, that corresponds to an angle where opposite is h and adjacent is 200 m.
4. Choose the ratio: tan(θ) = opposite/adjacent.
5. Solve:

tan(12.0°) = h / 200 m  h = 200 m · tan(12.0°)

Compute:

h ≈ 200 · 0.2126 ≈ 42.5 m

Answer: The cliff is approximately 42.5 m high.

Ladders Against Walls: Decide What the Given Length Represents

Ladder problems are classic because the ladder itself is a sloped distance. The wall is vertical, the ground is horizontal, and the ladder is the diagonal. The most common mistake is treating the ladder length as a vertical height or horizontal distance. It is neither; it is the sloped side.

Example 5: Ladder reach height (find height)

A 12.0 ft ladder leans against a vertical wall. The angle between the ladder and the ground is 70.0°. How high up the wall does the ladder reach?

Step-by-step

1. Triangle: ladder = diagonal (12.0 ft), height up the wall = h (vertical), ground distance from wall = horizontal.
2. Angle: at the ground where the ladder touches the ground: 70.0°.
3. Sides relative to the angle: opposite = h, hypotenuse = 12.0 ft.
4. Use sine: sin(70.0°) = h / 12.0 ft.
5. Solve:

h = 12.0 ft · sin(70.0°) ≈ 12.0 · 0.9397 ≈ 11.3 ft

Answer: The ladder reaches approximately 11.3 ft up the wall.

Choosing the Right Ratio Quickly (Mini Reference)

Unit discipline practice

For each problem you solve, write the final line in the format:

Answer: ________ (units)

If your calculator gives many decimals, round and keep the unit: 57.1 m, 11.3 ft, 8.6 m.