How to find the volume of a parallelepiped?

Can anyone help me find a formula to connect these values that relate to the volume of a sphere of n-dimension?

• in the 2nd dimension, n=2 and the coefficient of the volume equals 1 in the 4th dimension, n=4 and the coefficient of the volume equals 1/2 in the 6th dimension, n=6 and the coefficient of the volume equals 1/6 and I also need a 2nd equation for the odd numbers of the sequence in the 3rd dimension, n=3 and the coefficient of the volume equals 4/3 in the 5th dimension, n=5 and the coefficient of the volume equals 8/15 in the 7th dimension, n=7 and the coefficient of the volume equals 16/105 the equation for the evens and the odds will be different, i know that, and I also know that they need factorials but other than that i can not figure them out

• Answer:

  There is a single formula for the volume of an n-sphere that involves the gamma function. Alternatively, we can split by even/odd for simpler representations. Even: Each successive term in the even sequence is the next 1/(k!) term. The 8th dimension is 1/24 = 1/(4!). For even n, we have 1/((n/2)!) as the coefficient. Odd: This is bit trickier. Notice the numerator is always the next power of 2. The denominators are found by multiplying by the next odd number. 3*5 = 15. 15*7 = 105. So the 9th dimension coefficient is 32/945, because 32 is the next power of 2 and 105*9 = 945. In general, we have 2^((n+1)/2) / product(1,3,5,...,n) for odd n. There are couple edits above, it is now correct.