What is the length of the perimeter of mainland United States and that of Alaska
and Hawaii?

The CIA World Factbook lists the land boundaries of the United States as 12,248 
kilometers (7612 miles), and the coastline as 19,924 km (12,383 miles).  These 
numbers include all US territory.  The Factbook doesn't break these numbers 
down except to say that 2,477 km (1539 miles) of the land boundary consists of 
Alaska's border with Canada, and that 29 km (18 miles) consists of the border 
between Cuba and the US Naval Base at Guantanamo Bay.  The rest of the land 
boundary is presumably part of the perimeter of the contiguous 48 states.

http://www.cia.gov/cia/publications/factbook/geos/us.html

The Learning Network has a breakdown of coastline length by state.  According 
to their web page, Hawaii has 750 miles of coastline (which should be the sum 
of the perimeter of each of its islands).  Alaska has 5,580 miles of 
coastline.  That would make the total perimeter of Alaska 7,119 miles (5580 
miles of coast + 1539 miles of land boundary).  Subtracting Alaska and Hawaii, 
the contiguous 48 states have 6053 miles of coastline and 6055 miles of land 
boundaries, for a perimeter of 12108 miles.

http://www.infoplease.com/ipa/A0001801.html

Just the numbers I was looking for!

Using concepts from fractal geometry, mathematicians make the argument that the 
US coastline is of infinite length.  

Imagine two poles separated by 50 meters of rope.  If two men grab these poles, 
stretch out the line between them, and "walk off" the distance around a lake, 
they might find that it required 20 lengths of this rope to circumnavigate the 
lake.  Thus, they would calculate that the lake has 50 x 20 = 1000 meters of 
shore.

Now, if the length of the line is reduced to 20 meters, the men would be able 
to map the shore more accurately, as fewer rocky outcroppings would be skipped 
in the measuring process.  Using this shorter rope, the men might require 70 
lengths to go around the lake.  Thus, they would calculate the coast to be 20 x 
70 = 1400 meters in length.

Now, if the rope were 10 cm in length, the men would catch the individual rocks 
that make up the coast, but they would miss the pebbles.

If the rope were 1 cm in length, they would catch the individual rocks, but 
they would miss the pebbles.

If the rope were 1 mm in length, they would catch the pebbles, but they would 
miss the grooves in the edges of the pebbles.... and so on down to subatomic 
particles.

In short, the closer you trace the coastline, the longer it gets.

Some sources: 
  http://www.bbc.co.uk/dna/h2g2/alabaster/A425972
  http://www.msri.org/activities/jir/bwachtel/MeasuringtheCoast.html
  http://www.calresco.org/fractal.htm
  http://www.earthsky.com/2000/es001104.html

Google search:
  ://www.google.com/search?q=fractal+%22infinite%22+coastline

It appears that the answer above contains an error.  According to the
source referred to (http://www.infoplease.com/ipa/A0001801.html) the
coastline of Alaska is 6640 miles not 5580.  This is the total of
Alaska's Pacific coast (5580) and its Arctic coast (1060).  The answer
omitted the Arctic coast and the totals are off by this amount.