Question

how to calculate the height of a building using shadow length and angle of elevation

Answer 1

To calculate the height of a building using the shadow length and the angle of elevation, you can use basic trigonometry. Here's a step-by-step guide:

### Given:
- **Shadow Length (L)**: The length of the shadow cast by the building.
- **Angle of Elevation (θ)**: The angle formed by the line of sight from the tip of the shadow to the top of the building.

### Steps to Calculate the Height (H) of the Building:

1. **Identify the Triangle**:
- When you look at the scenario, you can form a right triangle where:
- The height of the building (H) is the opposite side.
- The length of the shadow (L) is the adjacent side.
- The line of sight from the top of the building to the tip of the shadow represents the hypotenuse.

2. **Use the Tangent Function**:
- The tangent of the angle of elevation (θ) is defined as the ratio of the opposite side to the adjacent side:
\[
\tan(θ) = \frac{H}{L}
\]

3. **Rearrange to find Height (H)**:
- To find the height of the building, rearrange the equation:
\[
H = L \cdot \tan(θ)
\]

4. **Calculate**:
- Plug in the values for shadow length (L) and angle of elevation (θ) into the formula you derived:
\[
H = L \cdot \tan(θ)
\]

### Example Calculation:
- Suppose the shadow length (L) is 10 meters, and the angle of elevation (θ) is 30 degrees.
- First, calculate \(\tan(30°)\):
\[
\tan(30°) = \frac{1}{\sqrt{3}} \approx 0.577
\]
- Now, calculate the height (H):
\[
H = 10 \cdot 0.577 \approx 5.77 \text{ meters}
\]

### Conclusion:
Now you have the height of the building based on the shadow length and angle of elevation. You can use this method for any values of shadow length and angles of elevation. Just ensure your angle is in a suitable format for the tangent function (degrees or radians) based on your calculator.