Question
height of building formula angle of elevation distance
Answers
GPT-4o mini
To find the height of a building using the angle of elevation and the distance from the observer to the building, you can use basic trigonometry, specifically the tangent function.
### Formula
Given:
- \( h \) = height of the building
- \( d \) = distance from the observer to the base of the building
- \( \theta \) = angle of elevation from the observer's position to the top of the building
The relationship can be expressed with the tangent function:
\[
\tan(\theta) = \frac{h}{d}
\]
To find the height of the building, rearrange the formula:
\[
h = d \cdot \tan(\theta)
\]
### Steps to Calculate the Height
1. Measure the distance \( d \) from your position to the base of the building.
2. Measure the angle of elevation \( \theta \) to the top of the building using a clinometer or similar device.
3. Substitute the values of \( d \) and \( \theta \) into the formula \( h = d \cdot \tan(\theta) \).
4. Solve for \( h \) to find the height of the building.
### Example
If you are standing 50 meters away from a building and the angle of elevation to the top of the building is \( 30^\circ \):
1. \( d = 50 \, \text{meters} \)
2. \( \theta = 30^\circ \)
Now calculate:
\[
h = 50 \cdot \tan(30^\circ)
\]
Using \( \tan(30^\circ) \approx 0.577 \):
\[
h = 50 \cdot 0.577 \approx 28.85 \, \text{meters}
\]
So, the height of the building would be approximately 28.85 meters.
### Formula
Given:
- \( h \) = height of the building
- \( d \) = distance from the observer to the base of the building
- \( \theta \) = angle of elevation from the observer's position to the top of the building
The relationship can be expressed with the tangent function:
\[
\tan(\theta) = \frac{h}{d}
\]
To find the height of the building, rearrange the formula:
\[
h = d \cdot \tan(\theta)
\]
### Steps to Calculate the Height
1. Measure the distance \( d \) from your position to the base of the building.
2. Measure the angle of elevation \( \theta \) to the top of the building using a clinometer or similar device.
3. Substitute the values of \( d \) and \( \theta \) into the formula \( h = d \cdot \tan(\theta) \).
4. Solve for \( h \) to find the height of the building.
### Example
If you are standing 50 meters away from a building and the angle of elevation to the top of the building is \( 30^\circ \):
1. \( d = 50 \, \text{meters} \)
2. \( \theta = 30^\circ \)
Now calculate:
\[
h = 50 \cdot \tan(30^\circ)
\]
Using \( \tan(30^\circ) \approx 0.577 \):
\[
h = 50 \cdot 0.577 \approx 28.85 \, \text{meters}
\]
So, the height of the building would be approximately 28.85 meters.