Library · 2. Core Claims · mark E

The diamond’s area

Claim

The void between four tangent circles has area (4 − π)r².

Construction

Four circles of radius r sit with their centres on the corners of a square of side 2r, so adjacent circles are tangent. At the middle of that square is a region touched by none of them, bounded by four concave arcs of radius r. That is the diamond.

Fig. 1 — four offcuts, right angle to right angle, close into the diamond.

Fig. 2 — the square on the four circle centres. Each circle puts one quarter disc inside it, and the four quarter discs come to one whole disc; what the square has left over is the diamond

Proof

Take the square whose corners are the four centres: side 2r, area 4r². Each circle contributes exactly one quarter-disc to the interior of that square, because the corner angle is 90°, and four quarter-discs make one whole disc, πr².

4r² − πr² = (4 − π)r² ≈ 0.8584r²

Each corner offcut from Claim A is a quarter of this, (4 − π)r²/4 ≈ 0.2146r², which is the smallest natural unit in the whole system.

Note

Aaron's slot.

Open questions

Whether the offcut deserves a name of its own in the project's vocabulary. It is the unit everything else is measured in, and it is currently called the offcut only here.

Checks

What	How	Source
Two routes, one number	Square minus circle here; four offcuts assembled in Claim A	reference/geometry-constants.md
Arithmetic	(4 − π) = 0.858407 and a quarter of it is 0.214602	reference/geometry-constants.md