Measuring a height you cannot reach: Pythagoras and angle
Why measure any other way than with a tape?
A gable, a vaulted ceiling, the ridge of a garden shed: the dimension exists, but you cannot reach it with a tape without putting up a scaffold. Indirect measurement replaces the ladder with two readings from the ground. Two distances and a right angle are enough.
How does Pythagoras measurement work?
The principle is the theorem itself. Stand some distance from the wall and aim at the high point: that is the hypotenuse. Then aim at the wall horizontally: that is the side along the ground. The height is the square root of the difference of the squares.
Example: from 6.00 m from the wall, the reading up to the top of the gable gives 7.50 m.
7.50² = 56.25 · 6.00² = 36.00 · 56.25 − 36.00 = 20.25 · √20.25 = 4.50 m
The instrument does that calculation on its own. What it does not do is guarantee that the second reading really was horizontal: that is where the measurement is lost.
By how much is the error amplified?
The stated ±2 mm on each reading do not stay ±2 mm in the result. In the example above, 2 mm on the hypotenuse moves the height by 3.3 mm, and 2 mm on the horizontal moves it by 2.7 mm — about 6 mm in total.
| Reading | Value | Effect of ±2 mm on the height |
|---|---|---|
| Hypotenuse | 7.50 m | ±3.3 mm |
| Horizontal | 6.00 m | ±2.7 mm |
| Calculated height | 4.50 m | about ±6 mm |
The amplification grows as the angle closes: the further you stand from the wall for the same height, the more alike the two readings become, and the more fragile their difference. So stand as close as the sightline allows.
What does angle measurement change?
The green 120 m model measures the angle on two axes, from −90° to +90°. One reading is then enough: the height is the distance multiplied by the sine of the angle. On the same gable, 7.50 m at 36.9° gives 7.50 × 0.600 = 4.50 m.
Two axes rather than one, because a hand-held instrument is almost always slightly tilted sideways: the second axis is what tells you so.
What else is the angle for, besides heights?
Reading a slope. The slope of a roof, a ramp or a drain is given in degrees or as a percentage, and both follow from the measured angle. One degree is about 1.7 %; 5° is 8.7 %. A roof at 30° therefore falls at 57 %.
Which model does what?
Both compute by Pythagoras. Only the green 120 m measures an angle, on two axes from −90° to +90°, which lets you read a slope directly. The 80 m keeps 99 readings in memory against 30 on the green 120 m, and states 80 metres of range against 120.
| Function | 80 m | 120 m green |
|---|---|---|
| Pythagoras measurement | yes | yes |
| Angle measurement | no | two axes, −90° to +90° |
| Stated reach | 80 m | 120 m |
| Memory | 99 readings | 30 readings |
See also: area and volume, the 80 m distance meter, distance meter or tape measure, and our laser distance meters.
- Pythagoras: height = √(hypotenuse² − horizontal²). At 6.00 m, a reading of 7.50 m gives 4.50 m.
- The ±2 mm on each reading become about ±6 mm on the height: indirect measurement amplifies the error.
- The further you stand from the wall for the same height, the more fragile the calculation.
- The green 120 m measures the angle on two axes, from −90° to +90°: one reading is enough.