Reuleaux triangle

The Reuleaux- triangle is obtained by the circuit simplest example of a orbiform ( "curve of constant width"). The distance of each point on one side from the opposite corner point is constant. The "constant width" is retained when rotating around the center of the area: The end point of the side is now the counterpoint of another side (the one that is bounded by the previous counterpoint). The Reuleaux triangle is named after Franz Reuleaux , a 19th century German engineer who pioneered the field of gear theory.

A Reuleaux triangle

To construct a Reuleaux triangle, you start with an equilateral triangle . A circle is drawn around each corner point that goes through the two opposite corner points. The intersection (i.e. the common area) of the three circles forms the Reuleaux triangle.

According to the Blaschke-Lebesgue theorem , the Reuleaux triangle has the smallest area of all uniform thickness.

The Reuleaux triangle can be generalized to regular polygons with 2 n + 1 sides. See curved polygon .

Area

When calculating the area of a Reuleaux triangle R , where the equilateral triangle ABC required for construction has the side length r , one first needs the area of the equilateral triangle, which is calculated as follows:

{\ displaystyle A _ {\ text {ABC}} = {\ frac {\ sqrt {3}} {4}} r ^ {2} \! \,}

.

In addition to the triangle ABC, a Reuleaux triangle consists of three equal segments of a circle that define the opening angle

{\ displaystyle \ theta = {\ frac {\ pi} {3}} \! \,}

possess (corresponding to one sixth full circle, ie 60 °) .
Thus, the area of a segment of a circle is A _s

{\ displaystyle {\ begin {aligned} A_ {s} & = {\ frac {1} {2}} r ^ {2} (\ theta - \ sin \ theta) \\ & = \ left ({\ frac { \ pi} {6}} - {\ frac {\ sqrt {3}} {4}} \ right) r ^ {2} \\\ end {aligned}} \! \,}

.

The area of the Reuleaux triangle is therefore:

{\ displaystyle {\ begin {aligned} A_ {R} & = 3A_ {s} + A _ {\ text {ABC}} \\ & = {\ frac {1} {2}} (\ pi - {\ sqrt { 3}}) r ^ {2} \\\ end {aligned}} \! \,}

.

scope

If the side length of the underlying triangle is ABC (equivalent to the width ), the circumference of the Reuleaux triangle is calculated ${\ displaystyle r}$ ${\ displaystyle U}$

{\ displaystyle U = \ pi r}

Three-dimensional generalization

A Reuleaux tetrahedron

The intersection of spheres with radius s , whose centers lie on the corners of a regular tetrahedron with side length s , is called Reuleaux's tetrahedron .

In contrast to the Reuleaux triangle, the diameters of the Reuleaux tetrahedron are not all of the same length. So the diameter that leads through the two points, which are located in the middle of two opposite edges of the body, is greater than s : their distance is