OFFSET
1,1
COMMENTS
The length of the legs of the k-th triangle of the Spiral of Theodorus are 1 and sqrt(k), while the length of the hypotenuse is sqrt(k + 1). Theodorus stopped at the 16th triangle, whose hypotenuse measures sqrt(17), because from the next triangle his spiral began to overlap (see A072895).
The area of the k-th triangle is (1/2) * sqrt(k), so the area A of the spiral constructed by Theodorus is A = (1/2)*Sum_{k=1..16} sqrt(k) = 22.234598300071...
The perimeter of the spiral of Theodorus (up to the 16th triangle) is 17 + sqrt(17).
LINKS
Wikipedia, Spiral of Theodorus.
FORMULA
Equals (1/2)*Sum_{k=1..16} sqrt(k).
EXAMPLE
22.234598300071301078545770187447994733070569...
MATHEMATICA
RealDigits[(1/2)*Sum[Sqrt[k], {k, 16}], 10, 120][[1]] (* Stefano Spezia, Jun 23 2024 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Gonzalo MartÃnez, Jun 23 2024
STATUS
approved