几何高度计算器

The Geometry Height Calculator finds the height of an object — a tree, building, flagpole, or mountain — using only a horizontal distance to its base and the angle of elevation to its top. The formula:

h = distance × tan(elevation angle)

This is one of the most practical applications of right-triangle trigonometry. You can measure heights you cannot climb. This tutorial covers the formula's derivation from SOHCAHTOA, three worked examples, and common applications.

The setup

Stand at a horizontal distance D from the base of an object. Look up at the top of the object. The angle from horizontal up to your line of sight is the angle of elevation (θ).

You, the base of the object, and the top of the object form a right triangle:

• The horizontal leg is the distance D.
• The vertical leg is the height h (what you want to find).
• The hypotenuse is your line of sight to the top.
• The right angle is at the base of the object (where it meets the ground).

Applying SOHCAHTOA

The angle of elevation θ has:

• OPPOSITE side: the height h.
• ADJACENT side: the distance D.

From TOA: tan(θ) = opposite / adjacent = h / D.

Solving for h: h = D × tan(θ).

Worked example 1 — finding tree height

You stand 30 meters from the base of a tree. The angle of elevation to the top is 35°. How tall is the tree?

h = 30 × tan(35°) ≈ 30 × 0.7002 ≈ 21.01 m.

So the tree is approximately 21 meters tall.

Worked example 2 — building from across a street

A surveyor stands 50 feet from a building. The angle of elevation to the roof is 60°. How tall is the building?

h = 50 × tan(60°) = 50 × √3 ≈ 50 × 1.732 ≈ 86.6 ft.

Worked example 3 — reversing the problem

If a 100 ft tall object is viewed from a distance of 200 ft, what is the angle of elevation?

From h = D × tan(θ): 100 = 200 × tan(θ) → tan(θ) = 0.5 → θ = arctan(0.5) ≈ 26.57°.

Adjusting for observer height

The basic formula assumes the observer's eye level is at the same height as the object's base. In practice, your eyes are about 1.5 m above the ground. To get the object's total height from ground level, ADD your eye-level height:

Object height = D × tan(θ) + observer eye-level height

For most rough applications, the eye-level correction is small compared to the building/tree height and is often ignored.

Angle of depression — the symmetric case

If you're ABOVE the object (looking down from a hilltop or building, say at a boat in the ocean), the "angle of depression" works analogously. The formula is mirror-image: you're looking down instead of up, but the same trig applies.

Real-world applications

• Forestry: measuring tree height for timber estimation. Clinometers measure the elevation angle directly.
• Surveying: determining building heights, bridge clearances, antenna heights.
• Navigation: sextants measure celestial angles for ship and aircraft positioning.
• Geology: measuring mountain or cliff heights from a known horizontal baseline.
• Sports: baseball broadcasting uses similar trig to estimate ball trajectories.
• Hunting / wildlife: rangefinders use this principle to compute distance to a target from its observed angular size.

Multiple measurements for accuracy

If you measure from two different distances D₁ and D₂ with elevation angles θ₁ and θ₂, you can derive both the object height AND distance via:

h = D₁ × tan(θ₁) = D₂ × tan(θ₂)

This redundancy lets you cross-check your measurements. If the two computed heights disagree significantly, one of your measurements is off.

Common mistakes

• Confusing elevation with depression. Elevation is angle UP from horizontal. Depression is angle DOWN. They're measured from the same baseline (horizontal) but go opposite directions.
• Using degrees in radian-mode calculator. tan(35°) ≈ 0.7. tan(35 radians) is a completely different value. Check calculator mode.
• Forgetting the horizontal distance must be ground-level horizontal. If the ground slopes or your distance is along a slant, the formula doesn't apply directly.
• Forgetting observer eye-level. For tall objects this is usually negligible, but for short objects (a 2 m hedge) eye-level adds ~75% error if ignored.