Question

A wire of diameter 1 mm has a resistance of 0.5 . What is the resistance of another wire of the same material and the same length but of diameter 0.5 mm?

Answer 1

The second wire has 1/2 the radius of the first wire.  

Now the area of a circle is (A =π r²) ... the area changes in proportion to the SQUARE of the circle's radius.

So if you cut both wires exactly perpendicular to their lengths, and look at the circular end where the cut is, the second wire has only 1/4 the cross-section AREA of the first wire.  Current in the second wire is trying to squeeze through a pipe that only has 1/4 the space to carry it.

The resistance of a piece of wire-2 is 4 times the resistance of a piece of wire-1 with the same length.

The resistance of wire-2 is (4 x 0.5 ohm) = 2 ohms .