Dart Board Mathematics
Guess Guess what is the fraction of the area of the middle ring to the whole area of the dartboard in the above two diagrams?
Solution Quite unexpected, your guess might be correct, even if you have not done any calculation! For diagram 1, Fraction of area of the middle ring to the whole dartboard For diagram 2, Fraction of area of the middle ring to the whole dartboard In general, this is true for any odd number of rings. If the radius of the outmost circle of the dartboard is denoted by r = 2n + 1, Fraction of the middle ring to the whole dartboard
Even number of rings Now, guess the fraction of the sum of areas the two middle rings of circles to the area of the whole dartboard with even number of rings (r = 2n is the radius of the outermost circle). Your guess might be even correct without calculation: Fraction of the sum of the area of the two middle rings to the whole dartboard
Final Guess Guess which area is larger : the area in yellow or the area in blue in the above two diagrams? Then do the calculations.