What it does
How tall is a tower you cannot climb? Stand some way off, measure the angle up to its top, and a right triangle does the rest. This page draws that triangle for you: the tower, the person, the line of sight and the angle, and then solves the problem step by step with tan θ = opposite / adjacent.
It covers the Class 10 topic some applications of trigonometry (heights and distances): the angle of elevation, the angle of depression, and the classic two-observer problems.
Learn the ratios first with the trig ratio explorer explorer and the theorem behind them in pythagoras proof playground. For measuring real trees in the field, see tree basal area.
The four kinds of problem
- One observer: a person stands d metres from the foot of a tower and sees the top at angle θ. Choose what to find: the height, the distance or the angle. The value being found is greyed out and filled in for you.
- Two, same side: two people in a line with the tower, s metres apart, see the top at θ (the nearer one) and φ (the farther one). The page finds the height and how far each person stands.
- Two, opposite sides: the tower stands between two people s metres apart.
- Looking down: someone at the top of a tower or cliff of height h sees an object on the ground at an angle of depression θ. The page finds how far away it is.
What each setting does
- d: distance along the ground from the person to the foot of the tower, in metres.
- h: height of the tower.
- θ and φ: angles in degrees, measured from the level line.
- s: the gap between the two people.
- e: eye height. Many textbook problems say the observer is, say, 1.5 m tall; the triangle starts at the eye, so e is added to the answer at the end.
A worked example
The page starts with one observer 30 m from a tower who sees the top at 60°.
- tan 60° = h / 30.
- h = 30 × tan 60° = 30 × 1.7321 ≈ 51.96 m.
Switch to Two, same side. The nearer person sees the top at 60°, the farther one at 30°, and they are 20 m apart. With x the nearer distance: h = x tan 60° = (x + 20) tan 30°, so x = 10 m and h = 10√3 ≈ 17.32 m. The farther person is 30 m away.
Looking down starts with a 75 m cliff and a boat seen at 30° below the level line. The angle of elevation from the boat is also 30° (alternate angles), so the distance is 75 / tan 30° ≈ 129.9 m.
Tips
- Always draw the picture first and mark the right angle. The page’s picture shows where it is.
- The angle of depression is measured down from the level line, not from the vertical.
- In two-observer problems, write two equations for the same h and solve them together.
Check yourself
Four fresh questions each time, using the special angles 30°, 45° and 60°. Use √3 ≈ 1.732. Type the answer, press Check, and open the worked answer if you need it. Nothing is saved.
FAQ
Why does the picture change scale? It zooms so the whole scene always fits your screen; while you drag, it stays still.
Is this from the NCERT book? It covers the same topic with fibub’s own problems and pictures. fibub is not part of NCERT.
Where the numbers come from
- Standard right-triangle trigonometry (tan θ = opposite / adjacent) applied to heights and distances. Problems and pictures written for fibub.
- Checked against hand-worked exact answers such as h = 10√3 m for two observers 20 m apart at 60° and 30°, and by putting every answer back into the angles.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.