It's not simple and sometimes impossible to divide a circle into n equal parts. That is, if the word "circle" signifies the border of a circular region, then, as was shown by the great Gauss, the construction with compass and ruler is only possible when Euler function φ is a power of 2: φ(n) = 2n. This happens exactly when n is the product of Fermat primes and a power of 2. Here are a few such numbers 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24 ... In particular Euclidian construction of a regular heptagon, i.e. a polygon with 7 sides is impossible.
However, if the question is of dividing the area of a circle, not the circumference, the problem is solvable in an elegant way that works for any
Remarkably, the length of each of the curves equals the length of the original semicircle. More Importantly, each of the regions bounded by two successive curves is exactly 1/7 of the given circle.
Let's check this in a general case of n parts. Take two curves: one on, say, k and n-k parts, the other on m and n-m parts, m >k. Assuming each part equals 2 so that k, n-k, m, and n-m are the radii of the four semicircles, the area enclosed by the two curves is half the sum
This is equal to
In case m = k + 1, i.e., when the region at hand is bounded by two successive curves the area is
But, in our notations, the radius of the big original circle is n/2 and the area is
which is exactly n times the area between two successive curves.
For an interactive construction with a Java applet, visit Divide a Circle into N Parts of Equal Area.