Library · 2. Core Claims · mark A

Area

Claim

Nuna's area is d² — the square on its own width.

Construction

Inscribe a circle of radius r in a square of side 2r. The circle touches each side once, at its midpoint, and what is left inside the square is four corner pieces, each bounded by two straight legs of length r and one quarter-circle arc of radius r. Call them the offcuts.

Bring the four offcuts together so their right-angle corners meet at a point, each turned a quarter from the last. The arcs now face outward and the figure closes: it is the curved diamond, the same void that sits between four tangent circles in the shape's own construction.

Nuna is one whole circle fused to one such diamond, so its area is the circle plus the diamond.

Fig. 1 — the square, its inscribed circle, and one of the four offcuts.

Proof

The square has area 4r² and the inscribed circle πr². The four offcuts are the remainder: 4r² − πr² = (4 − π)r².

The offcuts reassemble into the diamond with no gap and no overlap. This is an exact dissection, not an equality of measure, and the difference is worth insisting on: each offcut is an r × r cell with a quarter-disc removed at one corner, and four of them placed right-angle to right-angle produce a figure whose boundary is four concave quarter-arcs of radius r.

A = πr² + (4 − π)r² = 4r² = d²

The π cancels. A shape built entirely from circular arcs, every boundary point of it at distance r from some centre, encloses exactly the area of a square. Practically: a nuna tile and a square tile of the same width cover the same ground, so a floor quoted in square feet does not change when the shape does.

Fig. 2 — the dissection: the square of side 2r with its inscribed circle and four offcuts, the offcuts closed into the diamond, and the circle fused to the diamond. The areas add to 4r²

Note

Where were you when the π cancelled, and did you believe it at first?

Did you find this by algebra, or by cutting paper?

Open questions

Whether the dissection should be shown as an animation rather than a static figure. The claim is about rearrangement, and a still picture argues it weakly.

Checks

What	How	Source
Area of the exact tile path	Shoelace over 20,000 sampled points: 40,000.000 at r = 100, against a predicted 4r² = 40,000	_src/core.py
Dissection, independently	Square minus circle, and four offcuts assembled; both give (4 − π)r²	reference/geometry-constants.md
Earlier check on file	Shoelace on 4,000 points → 39,999.995	reference/geometry-constants.md
The dissection in the source	“Cutting the circle from the square and rearranging the corner pieces”	A New Shape, p. 9