What is the function to get the width/length of a house if I need a given ratio ($1:1.6$) and the outer wall has a thickness?

Question

We are dreaming building a house with a net (netto) area of $100m^2$.

Net means the thickness of the outer walls are not counted regarding the area.

Inner walls doesn't matter now.

The ratio needs to be $1 : 1.6$ regarding the outer sides.

The thickness of the outer wall is $40$ centimeters.

Example picture not proportional!: picture

The BIG question: what is the function for getting all the possible outer side lengths? The ones marked with "?" in the example picture.

From this, many would benefit, since "Olgyay", a bigger architect said passive houses in the temperate zone are the most energy-saving if their side ratio is 1:1.6

Answer 1

Let's say the shorter side of the house has length $x$ (on the outside). This means the longer side has length $1.6\cdot x$.

The inner rectangle, therefore, has sides of length $x-0.8$ and $1.6x - 0.8$, so the equation you want to solve is

$$(x-0.8)\cdot (1.6\cdot x - 0.8) = 100$$

This is a regular quadratic equation that can easily be solved using the standard technicque:

• Multiply out the left
• Bring everything to the left side
• Change the equation to the form $ax^2+bx+c=0$
• Remember that if $ax^2+bx+c=0$, then $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$