Calculating Total Length of Timber for Symmetrical Roof Truss
The image provides a diagram of a symmetrical triangular roof truss with a pitch of 35°. It's given that \( AC = 6 \) meters, and we need to calculate the total length of timber required to make the truss, considering all the lengths \( AD, FB, \) and \( BG \) are equal, as well as \( GC \).
Given the symmetry of the problem and the 35° pitch, we can break down the truss into different right-angled triangles and use trigonometric ratios to find the lengths of the sides.
For the main triangle \( ABC \):
- \( AC \) is the base and is given to be \( 6 \) meters.
- \( AB \) can be found using the given pitch angle.
The pitch of \( 35° \) tells us that angle \( BAC \) is \( 35° \). Since \( ABC \) is an isosceles triangle (because \( AB = BC \)), angle \( ABC = 35° \) as well.
In right-angled triangle \( ABD \):
- \( AD \) (which is half of \( AB \)) is the opposite side to angle \( BAC \) of \( 35° \).
- \( AC/2 \) or \( 3 \) meters is the adjacent side to angle \( BAC \) of \( 35° \).
Using the tangent function:
\[
\tan(35°) = \frac{AD}{3}
\]
Solving for \( AD \):
\[
AD = 3 \cdot \tan(35°)
\]
Since your truss is symmetrical and comprised of similar triangles, you can use this approach to calculate the total length of timber:
- Length of \( AD \) (same as \( FB \), \( BG \), \( GC \)) = \( 3 \cdot \tan(35°) \) meters
- Total length for all of these 4 members is \( 4 \cdot AD \).
In the smaller triangles \( AFD \), \( BFE \), and \( BGC \), the angle at \( F \) and \( G \) will still be \( 35° \). Given that each smaller triangle is half the size of the larger one, we can apply the same approach to find the length of \( DF \) (which will also equal \( FE \) and \( GB \)):
- \( DF \) (same as \( FE \), \( GB \); is half the size of \( AD \), so its length can be calculated as \( \frac{AD}{2} \)) = \( \frac{3 \cdot \tan(35°)}{2} \) meters.
- Total length for all of these 3 members is \( 3 \cdot DF \).
Now calculate the actual values for \( AD \) and \( DF \) using a calculator for tangent of \( 35° \), and then add them together to find the total length of timber needed:
\[
\text{Total timber length} = 4 \cdot AD + 3 \cdot DF
\]
Let's calculate the values using a calculator:
\[
AD = 3 \cdot \tan(35°) \approx 3 \cdot 0.7002 \approx 2.1006 \text{ meters}
\]
\[
DF = \frac{3 \cdot \tan(35°)}{2} \approx \frac{3 \cdot 0.7002}{2} \approx 1.0503 \text{ meters}
\]
Now, let's find the total length of timber:
\[
\text{Total timber length} = 4 \cdot (2.1006) + 3 \cdot (1.0503) \approx 4 \cdot 2.1006 + 3 \cdot 1.0503 \approx 8.4024 + 3.1509 = 11.5533 \text{ meters}
\]
So, the total length of timber required to make the truss is approximately \( 11.5533 \) meters, depending on how accurate the approximation or calculator is.
