Expert: Sherry Wallin - 5/8/2010

Question
QUESTION: Hi ,

I want to calculate the length of the ridge board for a roof. The total length is 6m and width is 5m. The pitch of the roof is 45 degrees. I have been able to calculate the following:

1/2 of 5m

Common Rafter  = ½ span (2.5m) x Secant 45 degrees (1.41421) = 3.53525


To calculate hypotonus for ridge rafter:

Hyp = √adj sq + opp sq

3.535525 sq = √2.5 sq + opp sq

3.535525 sq – 2.5 sq = opp sq = 12.50 – 6.25 = opp

Opp = √6.25 = 2.5m

ANSWER: It took me a moment to figure out what it is that you don't know. Basically you have a right triangle and you know the bottom leg is 2.5m and you want to find the hypotenuse. So you have:
cos 45' = 2.5/x where x is the length of the unknown hypotenuse. If you rewrite this relationship and multiply x on both sides you will have x*cos 45' = 2.5 BUT this is not what you want, you want just x so now you must divide by cos 45' on both sides resulting in x = 2.5/cos 45'. What you might be concerned about is that sec 45' = 1/cos 45' so instead, x = 2.5*sec 45'. The rest of what you are doing appears to be using Pythagorus's theorem a^2+b^2 = c^2. You can't take the square root of both sides on the individual terms. When you take the square root of the 3.53525 you need to take the square root of [(2.5)^2 + opp^2], i.e. the square root of both terms. This is a lot of unnecessary work since when you have a right triangle and one 45' angle the other angle has to be 45' and that means both the legs will be equal, i.e., hence this is why you arrived that the opposite was 2.5m also. One final observation or technique to tuck into your belt:
If you are using the angle the ridge board makes with the hypotenuse, you could find the hypotenuse by taking the cos(angle) = adj/hyp and solve =for the hypotenuse and then you could use the sin(angle) = opp/hyp. You found the hyp in the previous calculation so you could find the opp directly:
sin 45' = opp/3.5.525 so opp = 3.53525*sin 45'

Math Prof

---------- FOLLOW-UP ----------

QUESTION: Just to make sure that you have understood me correctly as i dont think i explained myself well i have drawn the picture of the roof for you to see. I hope that i have calculated it correctly. it has been many years since i last done maths so forgive me:)

Answer
Ok, hopefully this makes sense to you. The hypotenuse of the roof pitch is as you calculated 3.536, this is the leg of the right triangle that you have for your hip rafter (the base leg), the other leg is 2.5m (the rise) so the hypotenuse of that is your hip rafter, i.e., 3.536^2 + 2.5^2 = x^2 and then take the square root of both sides and you will get 4.33 for the length of the hip rafter.

Math Prof