Hipped Roof Span Calculation

This solution determines the span of a hipped roof using the given slope angle and height.

Geometric Interpretation

Visualize a right-angled triangle formed by the roof's height, half of the room's span, and the roof surface itself. The slope angle is the angle between the horizontal plane (half-span) and the roof surface.

• Height (h) = 3 m
• Slope Angle ($\theta$) = 60 degrees
• Half Span = w
• Full Span = S = 2w

Using Trigonometry

The relationship between the angle, height (opposite side), and half-span (adjacent side) in the right-angled triangle is given by the tangent function:

$ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{w} $

Calculation Steps

1. Substitute the known values into the formula:

   $ \tan(60^\circ) = \frac{3 \, \text{m}}{w} $

2. Solve for w (half-span):

   $ w = \frac{3 \, \text{m}}{\tan(60^\circ)} $

3. Use the known value $\tan(60^\circ) = \sqrt{3}$:

   $ w = \frac{3}{\sqrt{3}} \, \text{m} $

   $ w = \frac{3\sqrt{3}}{3} \, \text{m} = \sqrt{3} \, \text{m} $

4. Calculate the numerical value for w:

   $ w \approx 1.732 \, \text{m} $

5. Calculate the full span (S) by multiplying half-span by 2:

   $ S = 2 \times w = 2 \times \sqrt{3} \, \text{m} $

   $ S \approx 2 \times 1.732 \, \text{m} \approx 3.464 \, \text{m} $

Conclusion

The calculated span is approximately $3.464$ m. This value falls between $3.4$ m and $3.6$ m, confirming the expected range for the room's span.