Why manholes are round? (Weird Insights)

The ever-evident structure is found almost everywhere and the place where some comic villains usually dwell.

That's it. Manholes serve an essential purpose, as they can cover the sewers for animals or people falling into them.

However, most manholes carry a round shape. But why is that?

One might wonder that these manholes are easy to align in a uniform position at any horizontal angle due to their round shape.

But, there lies another advantage of such round manholes.

Let us take some help from mathematics to understand the solution to our manhole problem, shall we?

The problem

Placing a square in-between a pair of parallel lines and rotating it horizontally will distance the lines to increase.

However, if we place a circle to rotate it from the centre of those two lines, the length between those lines remains unchanged.

The shape of the circle forms it as a curve of constant width.

Now, let us divulge a little bit into the basic concepts of geometry.
 

The curve width at any particular direction is the distance that lies between the pair of perpendicular supporting lines in that direction.

A supporting line has a single common point on a straight line with its supporting curve, which falls on either side of that line.

Cutting by the perimeter on the shaded region, the resulting curve is called the Reuleaux triangle (spelt like the Roo-Low triangle) after the German mechanical engineer Franz Reuleaux (1829 – 1905).

The Reuleaux triangle consists of a curve with uniform width enabling it to maintain a constant linear distance during rotation.

Any regular polygon carrying sides of an odd number can help to form a Reuleaux polygon.

The rolling of one reuleaux polygon alongside the boundary of a similar polygon, the shape that comes from its centre should give rise to another reuleaux polygon. 
 

To obtain the perimeter of a Reuleaux polygon, use the formula 

Perimeter = Π x Diameter

The width of the polygon stands as the diameter.

The French mathematician and astronomer - Joseph-Emile Barbier (1839-1889) proved the theorem in 1860.

Several Reuleaux polygons with a similar width will carry an equal perimeter.

Reuleaux polygons with the same width shall have an area of the triangle as a minimum and the circle as maximum.

A wheel carrying a shape of a Reuleaux triangle should cover an equal distance in a single rotation, similar to a circular wheel with the same width.

The reduced area will help to save some material in the Reuleaux triangle, which is wasted on a circle.

However, since the centre of the Reuleaux triangle constantly moves, unlike a circle, during rotation, the shape is not used for making wheels. 

The solution

The rotation of a Reuleaux triangle along its centerline ends up forming a Reuleaux tetrahedron.

These sides of this shape will carry curved surfaces.

Similar to Reuleaux triangles, Reuleaux polyhedrons can separate a pair of planar surfaces by equal distances. 

Therefore, a curve carrying the same width can never slip within.

However, if the lid of a manhole consists varying width, it might slip in by aligning itself with the broader portion of the hole.

A constant-width curve does not slip in.