Visual Angle of Elevation, Depression & Sightline Rangefinder
The angle of elevation $\theta = \arctan(\text{Rise} / \text{Run})$ equals the alternate-interior angle of depression from the target back to the observer, with line-of-sight range $L = \sqrt{\text{Run}^2 + \text{Rise}^2}$ and percentage slope $\text{Grade} = 100 \times \tan\theta$.
θ = arctan(Rise / Run), Slope % = 100 × (Rise / Run)Inputs & Parameters (2)
20-Row Comparison Table
| # | Horizontal Ground Distance (Run) (m) | Hypotenuse (c) | Angle α | Area (½ab) |
|---|---|---|---|---|
| 01 | 2 m | 24.083 u | 85.24° | 24 u² |
| 02 | 7 m | 25 u | 73.74° | 84 u² |
| 03 | 12 m | 26.833 u | 63.43° | 144 u² |
| 04 | 17 m | 29.411 u | 54.69° | 204 u² |
| 05 | 23 m | 33.242 u | 46.22° | 276 u² |
| 06 | 28 m | 36.878 u | 40.6° | 336 u² |
| 07 | 33 m | 40.804 u | 36.03° | 396 u² |
| 08 | 38 m | 44.944 u | 32.28° | 456 u² |
| 09 | 43 m | 49.244 u | 29.17° | 516 u² |
| 10 | 48 m | 53.666 u | 26.57° | 576 u² |
| 11 | 54 m | 59.093 u | 23.96° | 648 u² |
| 12 | 59 m | 63.695 u | 22.14° | 708 u² |
| 13 | 64 m | 68.352 u | 20.56° | 768 u² |
| 14 | 69 m | 73.055 u | 19.18° | 828 u² |
| 15 | 74 m | 77.795 u | 17.97° | 888 u² |
| 16 | 79 m | 82.565 u | 16.9° | 948 u² |
| 17 | 85 m | 88.323 u | 15.77° | 1,020 u² |
| 18 | 90 m | 93.145 u | 14.93° | 1,080 u² |
| 19 | 95 m | 97.985 u | 14.18° | 1,140 u² |
| 20 | 100 m | 102.84 u | 13.5° | 1,200 u² |
Key Analytical Takeaways
- 01By parallel horizontal sightlines, the upward Angle of Elevation from observer to summit equals the downward Angle of Depression from summit to observer.
- 02A $45^\circ$ angle of elevation corresponds to a $100\%$ civil engineering grade ($\text{Rise} = \text{Run}$).
- 03Doubling horizontal distance does not halve the angle $\theta$; instead it halves $\tan\theta$.
Frequently Asked Questions
Why is a 100% slope equal to 45 degrees and not 90 degrees?+
Percent grade is defined as 100 × (vertical rise / horizontal run). When rise equals run, the ratio is 1.0 (100%), which forms an isosceles right triangle with a 45° angle.
How do surveyors find building height using an inclinometer?+
They measure horizontal baseline distance d from the structure and sight the roof angle θ, computing height as H = d·tan(θ) + eye_height.
Why is the angle of elevation equal to the angle of depression?+
Because horizontal sightlines at the observer's eye level and the elevated target's level are parallel, the line of sight forms alternate interior angles of identical measure θ.