Heights and Distances Calculator

Solve heights and distances problems with angles of elevation: find a height, a distance or an angle, or a height from two observation points, with exact √3 answers for 30°, 45° and 60°.

Find	Height, from the distance and the angle of elevation
Distance from the foot (m)	20
Angle of elevation (°)	60
Observer's eye height (m, optional)	0

Find

Distance from the foot (m)

Angle of elevation (°)

30°, 45° and 60° give exact answers with √3.

Observer's eye height (m, optional)

Added to the height when the angle is measured from eye level.

Height

20√3 m ≈ 34.64 m

With a standard angle the answer is exact: 20√3 m, about 34.641 m (using √3 ≈ 1.732, 34.64 m).

Exact surd. Checked numerically.

Method: tan θ = opposite ÷ adjacent
Worked out with: AllCalcHub exact surds (p + q√3)
Check: Every step compared at several values of x

The right triangle

About the Heights and Distances Calculator

The angle of elevation is the angle between the horizontal and your line of sight up to the top of an object. With a vertical object on level ground, the height, the distance and the line of sight form a right triangle, so tan θ = height ÷ distance.

With the angles 30°, 45° and 60° the calculator keeps √3 exact, so you see answers like 20√3 m as well as the decimal.

The formula

$\tan\theta = \frac{\text{height}}{\text{distance}}$

Tangents of the standard angles

θ	tan θ
30°	1/√3 = √3/3
45°	1
60°	√3

Worked example

From 20 m away, the angle of elevation is 60°

h = 20 × tan 60° = 20√3
≈ 34.64 m

Questions people ask

What's the angle of depression?

Looking down from a height, the angle below the horizontal. It equals the angle of elevation from the other end (alternate angles), so use the same calculator.

Why isn't my answer a whole number?

tan 30° and tan 60° involve √3, so answers are often multiples of √3. Textbooks often take √3 ≈ 1.732.