why we use the circular sewage lids instead of anothe shape?give me some mathematical explanation.
If you think about it for a moment, in order for the cover to not be able to fit through the smaller sized hole in any vertical orientation, the widths of both the cover and the hole must be constant, the hole slightly smaller than the cover. With the hole being slightly less than the cover by a suitable amount of overlap, no matter what orientation the cover is placed in relative to the hole, the hole opening is always smaller than the cover width. What is the one shape that can satisfy this requirement? Of course!. The circle. The circular cover is always faced with trying to fit its diameter through a hole that is, by design and necessity, slightly smaller in diameter than the cover diameter. How ingeneous.
That said, many people would be convinced that all manhole covers must be round to prevent the cover from falling through the matching opening. Having led you down this garden path, I must now tell you that any convex curved shape of constant width can provide the same capability. Of course, your next queastion is obviously what other closed curve shapes are of constant width? Many, surprisingly.
First let us address the question of what is a constant width curve? The circle is the simplest closed convex curve having a constant width as defined by its diameter. More generally, a closed convex curve is one that makes contact with all four sides of a square when rotated within the square. The constant width of the closed curve is therefore defined as the distance between either of the two sets of parallel lines forming the square. Constant width curves have the same width regardless of their orientation within the square.
Surprisingly, there are an infinite number of closed convex sahpes that will meet this requirement, though admittedly, not very different in shape. The "circle" and the "Reuleaux triangle" define the boundries of possible shapes. The circle is the largest area shape while the Reuleaux triangle is the smallest area shape for a given maximum width. The Reuleaux triangle is defined as follows.
1--Draw an equilateral triangle with sides equal to "r".
2--With a vertex as center, swing an arc of radius "r" between the other two vertices.
3--Repeat this at the other two vertices and you will have an equilateral triangle with three equal length, equal radius curved legs.
4--This shape has a constant width of "r".
Consequently, a manhole cover of this shape will never fall through the hole it is covering as the width of the hole is always smaller than the cover by the typical overlap.
A round cover with outer diameter (width) of 3 ft. has an area of 7.068 sq.ft. while a Reuleaux triangle shaped cover with the same outer diamater (width) of 3 ft. has an area of 6.343 sq.ft. Granted, there is not that much difference in area and, realistically, the only significance to what we have learned here is that a manhole, and its cover, need not necessarily be round.
Between the circle and the Reuleaux triangle lie an infinite variety of constant width shapes built up around acute triangles. I doubt if any of them would make it worthwhile building them in the non-round shapes and, being even more realistic, it is probably cheaper to manufacture purely round manholes than it is to produce custom made non-round ones.
To illustrate that the shape does not have to be based on an equilateral triangle, consider a randomly derived acute triangle ABC with vertices on a circle, A left, B above and C right opposite A. If you draw this yourself, be sure it is not equilateral.
Extend both ends of all three sides beyond their vertices.
Using point A as a center, swing an arc of some random radius from p1 on BA extended to p2 on CA extended.
Using point C as a center, swing an arc with radius Cp2 from p2 to p3 on CB extended.
Using point B as a center, swing an arc with radius Bp3 from p3 to p4 on AB extended.
Using point A as a center, swing an arc with radius Ap4 from p4 to p5 on AC extended.
Using point C as a center, swing an arc with radius Cp5 from p5 to p6 on BC extended.
Using point B as a center, swing an arc with radius Bp6 from p6 to p1 on BA extended.
Any pseudo maximum diameter passing through any one of the three vertices measures exactly the same, making the randomly derived shape constant width, and capable of being the shape of a manhole and its cover.