How can dimensionless points give rise to a line with length? A line is a collection of infinitely many points. By definition, a point has no dimensions. But, how can infinitely many dimensionless points give rise to a line with a dimension. This is the same case with planes, solids and higher dimensions too...
Thanks in advance for any help..!!
 A: "Being dimensionless" has never been the definition of the point. Euclid attempted to define "the point" like that. But he never used that definition to define other things or to prove theorems. It turns out that the point does not have a definition. Most of the pseudo paradoxes use some sort of definitions in the case of objects that are not even defined.

What is the definition of the point if it is an undefined term?  The definition of the point as an undefined term emanates from the axioms of geometry which, loosely speaking, establish the relationships between undefined terms like the point.
The best example is the Veblen's axiom system of geometry. If you read
this answer to a question you will do a fun exercise related to defining undefined stuff. 

(In theory, it is possible to establish a system of axioms that use other undefined terms but the point. This way the point becomes a defined concept. For instance Hilbert's axiom system handle the term straight line as an undefined term. The Veblen system defines the concept of line based on the undefined concept of the point.)

The best example is Veblen's axiom system of geometry. Read this  answer to a question. It'll be fun to sense the way the intuitive concept emerging  from nothing.
If something is paradoxical then this is paradoxical: Undefined concepts serve as the basis of definitions. How come that knowledge is possible at all?

So points exist based on the axioms of geometry without even mentioning their size. They do not have a size at this level.
Then come the axioms of congruence plowing the field so that the concept of length (an angle) can be grown. (And then there is Archimedes' axiom. ) 
At this level the concept of length is established. And then it turns out that the poor points do have zero length as their size can be defined as the limit the size of shrinking intervals... If you get to the concept of length via the axiomatic development of geometry then you wouldn't even feel the need to ask the question about the dimensionless objects forming objects that have dimensions.

Another interesting example to experience the weakness of our everyday intuition is the field of Conway's surreal numbers. The geometry of the surreal line is very interesting. So to speak, inside of our points whole new worlds are hiding (If you like.) But, this has to be enough for the time being. 
A: Try this theory on for size:
Consider two distinct points on a line.  There must be some distance separating them.  Obviously, it cannot be a minimum distance - we can always move the points closer together.  So there is no finite amount of distance needed to separate them.  But does this mean the distance between them can be reduced to zero?  NO - then the two points wouldn't be distinct anymore!  So any two points on a line, if truly distinct, must be separated by at least an infinitesimal amount.
SO: Although the infinite points on a line do not add up to any length, all of the infinitesimal spans needed to keep the points distinct DO add up to some length.
