What is the area of a treble and the area of a double?
First, I should comment that there is no single precise answer to your question because the size of the Trebles and Doubles areas depend on the construction of the dartboard, whether wires or blades are used to separate segments and the specifications of the equipment used to manufacture the dartboard which may vary a little from one manufacturer to another. In addition, the Doubles and Triples areas are bounded by circles and the calculation of the area of a circle involves the value of pi, which is a transcendental irrational number, meaning that it cannot be calculated exactly.
I found the following statement published on the web page http://www.ehow.com/about_5055012_regulation-dart-board-dimensions.html:
"The double number zone area should measure 0.773 square inches, and the triple number zone area should measure 0.479 square inches."
But the site does not quote the method for arriving at this answer. So, I have constructed the following method for calculating these numbers and judging the correctness of these estimates.
According to The World Darts Federation web site http://www.dartswdf.com/wp-content/uploads/2010/12/WDF-Playing-Tournament-Rules.
The width of Double and Treble dimensions are, for conventional wire boards, measured inside to inside = 8.0mm or .315 in. and for boards manufactured with strip material, also known as "Blade" boards, measured apex to apex = 10.00mm or .39 in.
Also from the World Darts Federation, the radius of the outside of the Doubles Ring, that is the distance from the Outside edge of 'Double' wire to Centre Bull = 170.0 mm or 6.69 in., and the radius of the outside of the Trebles Ring, that is the distance from the Outside edge of 'Treble' wire to Centre Bull = 107.4 mm or 4.23 in. Using the .39 in. for the width of the Doubles and Trebles rings, the radius of the inside Doubles Ring would be 6.3 in. and the radius of the inside Trebles Ring would be 3.84 in.
My reasoning using the standard definition for the area of a circle, Area = pi (3.14159) times the radius squared, is that the outside and inside of the Doubles and Trebles rings describe disks that can be used to calculate the area of the Doubles and Triples Rings.
I say the area of the Triples Ring is equal to the difference between the area of the disk with radius 4.23 in. and the disk with radius 3.84 in. and the area of any one segment of the Triples Ring is equal to 1/20th of the area of the total Triples Ring.
Similarly, the area of the Doubles Ring is equal to the difference between the area of the disk with radius 6.69 in. and the disk with radius 6.3 in. and the area of any one segment of the Doubles Ring is equal to 1/20th of the total Doubles Ring.
So, the area of the Triples Ring is equal to 3.14159 times 4.23 squared minus 3.14159 times 3.84 squared, or 9.98 sq. in. and the area of one segment (1/20th) is 0.50 sq. in.
And, the area of the Doubles Ring is equal to 3.14159 times 6.69 squared minus 3.14159 times 6.3 squared, or 16.07 sq. in. and the area of one segment (1/20th) is 0.80 sq. in.
I have arrived at .50 sq. in. for the Trebles area and .80 sq. in. for the Doubles area which are not identical but not all that different to the numbers quoted from ehow.com, .479 and .773 respectively.