I love the Reuleaux triangle so much. It's used in industry to drill squared holes! How crazy is that!!

Solution:

So, each segment s of the ABC inscribed triangle is 6cm because of how the Reuleaux triangle is composed. That allows us to get the center position p from any vertex, for it's equal to 2/3 of the triangle's height h, thus:

So, the radius r of the inscribed circle is:

So in numbers:

The inscribed equilateral triangle has sides of length 6.

The perpendicular has length 5, because Pythagoras 3^2 + 5^2 = 6^2.

The extension of the perpendicular must be 1, because 5 + 1 = 6.

Platon says the sides of an equilateral triangle's half have ratio 1:2:\sqrt{3}.

Therefore, the radius of the inscribed circle is \frac{3}{\sqrt{3}}+1 \approx 2.732.

I thought of a way to answer the puzzles without showing the answer. We can use 'SN's private messages' to send the answer to you. The stacker who answers you privately leaves a comment on the post saying that they've answered.

wallet > withdraw > lightning address

Update:

Reasoned answer for previous iteration: see #746568

Fun fact about Reuleaux triangles: they have a constant width, meaning that for every pair of parallel supporting lines (two lines of the same slope that both touch the shape without crossing through it) the two lines have the same Euclidean distance from each other, regardless of the orientation of these lines, leading to this kind of surprising behavior (short YT clip)