# How do I calculate the inclined area of a roof?

I think I already know the formula, but wanted to check if there are any instances where this may not work. I have been estimating construction costs for 5 years and haven’t found one instance where the following formula doesn’t give the correct answer:

Incline area = flat area (measured on plan)/cos (pitch in degrees)

So far this has worked for:

• Square, rectangular, hexagonal, circular, curved edge roofs
• Roofs of varying pitches eg. 5 degrees one side, 10 degrees the other. Just measure 2 separate areas and use the formula

It wouldn’t work for: - Roofs that are arced or domed vertically

Can anyone prove me wrong because I’ve seen so many people losing their mind over something so simple?

• For spherical domed roof, if the pitch is $\theta$ at the edge, the surface area is given by 2$\pi r^2 (1- cos(\theta))$. See here for more details: en.wikipedia.org/wiki/Spherical_cap where r is the radius of the dome. If you have the plan view radius (a) instead, you can use $r = \frac{a}{cos(\theta)}$ as you are already doing before to get radius. – user625 Jun 21 '18 at 12:43
• For an arced dome, it is a portion of a cylinder. So, you can use $2r\theta \cdot$ length where $r$ is the radius of the cylinder. Again, you can convert radius $r$ into plan view width by using $r = \frac{a}{cos(\theta)}$ – user625 Jun 21 '18 at 12:50

## 1 Answer

Just combining the comments above as an answer.

1. Spherical domed roof (shaped like a cap):

$\theta$ is the angle of the roof with the horizontal at the edge

$a$ = plan view radius

Area = $2\pi\cdot a^2\cdot\frac{1-\cos(\theta)}{\cos(\theta)}$

1. Arced dome (portion of a cylinder):

$\theta$ is the angle of the roof with the horizontal at the edge in radians

$a$ = plan view width

$l$ = plan view length

Area = $2\theta\cdot\frac{al}{\cos(\theta)}$