Calculate the volume of the roof

What you need:

• Paper and pencil
• Possibly. Yardstick
• calculator

A gable roof is a prism

• The roofs of simple houses are in most cases so-called. Gable roofs. They consist of two rectangles on the sides, which form the roof ridge in an inclined position with one long side. An equilateral triangle is created on the front (photo 1).
• The geometric figure of such a gable roof is a prism - known from the typical Toblerone box, which in this case is a roof that lies across.
• The base of such a prism is a triangle, the height of the prism is the length of the roof rectangle.

Calculate the volume - this is how you proceed

Which sizes are known? For a real task on a house, in most cases you will know the roof pitch, i.e. the roof pitch angle α in the front triangle. In addition, the lengths of the roof (ridge length) l and the sloping side (rafter length) s will be known (cf. Photo 1). For the calculation, assume a roof pitch of α = 25 °, a roof length of l = 12 m and a slope s = 3 m:

1. The volume of a prism is easy to calculate. The following applies: V = base area x height. As already explained, the base of the gable roof is a triangle; the height of the prism is the roof or Ridge length; thus V = triangular area x ridge length.


Prism calculation - how to calculate the volume

The volume of many geometric bodies can be relatively ...

2. So if you want to calculate the volume of your roof, you first need to know the area of ​​the roof triangle on the front. For the area of ​​a triangle, F = 1/2 base side x triangle height (1/2 g x h as a formula).
3. Unfortunately, you do not know the base side g or the triangular height h of the triangle, but only the base angle α and the inclined side s. School geometry helps here, because the following applies: sin α = h / s and cos α = 1/2 g / s. From this you calculate h = s * sin α = 3 m * sin 25 ° = 1.27 m and g = 2s * cos α = 6 m * cos 25 ° = 5.44 m (you could also have used tan or Pythagoras can).
4. For the area of ​​the front triangle, these two sizes give you F = g x h = 5.44 m x 1.27 m = 6.91 m² (all values ​​rounded to two places behind the decimal point).
5. The following then applies to the volume of the roof: V = triangular area x ridge length = 6.91 m² x 12 m = 82.92 m³.