Construction
With a compass, sweep an arc sufficient to enclose the desired figure. With radius unchanged, sweep a sufficient arc centred at a point on the first arc to intersect that arc. With the same radius and the centre at that intersection sweep a third arc to intersect the other arcs. The result is a curve of constant width.
Equivalently, given an equilateral triangle T of side length s, take the boundary of the intersection of the disks with radius s centered at the vertices of T.
By the Blaschke–Lebesgue theorem, the Reuleaux triangle has the least area of any curve of given constant width. This area is, where s is the constant width. The existence of Reuleaux polygons shows that diameter measurements alone cannot verify that an object has a circular cross-section.
The area of Reuleaux triangle is smaller than that of the disk of the same width (i.e. diameter); the area of such a disk is .
Read more about this topic: Reuleaux Triangle
Famous quotes containing the word construction:
“No real “vital” character in fiction is altogether a conscious construction of the author. On the contrary, it may be a sort of parasitic growth upon the author’s personality, developing by internal necessity as much as by external addition.”
—T.S. (Thomas Stearns)
“There’s no art
To find the mind’s construction in the face.”
—William Shakespeare (1564–1616)
“No construction stiff working overtime takes more stress and straining than we did just to stay high.”
—Gus Van Sant, U.S. screenwriter and director, and Dan Yost. Bob Hughes (Matt Dillon)