First We can start by drawing a diagram of a regular octagon.

Since the octagon is regular, each of its eight interior angles is 135 degrees (we can determine this by dividing the 360 degrees of a full circle by the 8 angles). We can then draw a line from the center of the circle to one of the vertices of the octagon, which will bisect one of the angles into two 67.5 degree angles. From there, we can use trigonometry to find the length of one of the sides of the octagon:

sin(67.5) = side/10

side = 10 x sin(67.5)

side ≈ 8.6603 cm

Now that we know the length of each side of the octagon, we can find the area by dividing the octagon into 8 congruent isosceles triangles and then summing their areas. Each of these triangles has a base of length 8.6603 cm and a height equal to half the radius of the circle (5 cm).

Therefore, each triangle has an area of 1/2 x 8.6603 x 5 = 21.651 cm^2. Since there are 8 such triangles, the area of the octagon is 8 x 21.651 = 173.21 cm^2.