Squaring Octagon house

An octagon iinscribed in a square

 

geometry - regular octagon with inscribed square - area ratio question - Mathematics Stack Exchange

Part 1: Verifying the "20% Rule"

If you have a fixed amount of exterior wall material (say, 100 feet of wall) and you build a square, you get 625 square feet of floor space. If you take that exact same 100 feet of wall and fold it into a regular octagon, you get roughly 754 square feet of floor space.

• Result: You get ~20.7% more floor space using the exact same amount of wall materials, insulation, and siding.

Part 2: Fixed Corners

What happens if we keep the four corners of the square (A, C, E, G) locked in place, and we "pull" the walls outward to form a perfect, regular octagon?

If we assume the shape in your image becomes a regular octagon (where all 8 sides are equal lengths, and all angles are 135 degrees), here is the exact geometrical math:

1. The Perimeter Increase:

• The math: The perimeter of the octagon compared to the square is a ratio of 

  $\frac{2}{\sqrt{2+\sqrt{2}}}$

  .

• The Result: The total length of the exterior walls only increases by 8.2%.

2. The Floor Surface (Area) Increase:

• The math: The area of the octagon compared to the inscribed square is exactly 

  $\sqrt{2}$

  .

• The Result: The floor space increases by a massive 41.4%.

The Real-World Example (For your AI Estimator)

Let's plug real numbers into this so you can see why this is so economically powerful for your software.

Imagine your core square (A-C-E-G) is 24x24 feet.

• The Square House:

  • Floor Space = 576 sq ft.

  • Exterior Wall = 96 linear feet.

• The Octagon House (pulling the 4 walls out into triangles):

  • To make a perfect octagon, those triangular points (B, D, F, H) stick out exactly 5 feet past the square.

  • Floor Space = 814 sq ft. (You gained 238 sq ft of living space!)

  • Exterior Wall = 104 linear feet. (You only added 8 total feet of wall!)

The Octagon House - 99% Invisible