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I’m looking to convert my attic to a home cinema, and I’m trying to work out what the largest size screen I can have in there (taking the roof pitch into account) and I’m scratching my head at what the equation will be...

Initially I’m going on rough dimensions as an example:

Triangle base is 5.6 meters in width, the highest point of the triangle is 2.25m.

The centre of the triangle angle is 102.4 degrees and the bottom left and right angles are 38.8 degrees each.

A screen rectangle is always of 16:9 ratio or 1:1.78

So I guess I’m trying to work out what the largest heigh / width of the screen (rectangle) possible within this triangle. As the dimensions are rough right now and will probably change due to available screen sizes, insulation and plastering and what not, it would be handy to know the working to this if possible so I can adjust as necessary.

Hope this makes sense! Any help would be greatly appreciated.

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The drawing below is at $2:1$ scale. Triangle ABC is your end wall. DE has a slope of $\frac 98$ and represents the corner of a TV with its center bottom sitting on the floor in the middle of the wall. The biggest screen that fits is rectangle GFHI, which is $1.32$ meters high and $2.33$ meters wide. The diagonal is about $2.678$ meters, or $105$ inches. The problem with this approach is that TVs often sit on a base so assuming we just need room for a $16:9$ rectangle is not correct. They often also have rounded corners, so you get a little more room.

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