I do not know exactly what you mean, but here is one way.Originally Posted byhari-kj

Construct the regular pentagon such that one of the 5 equal sides is horizontal.

Let us say also that this horizontal side is the top or topmost of the pentagon, so the bottom or bottom most is a vertex or corner of the pentagon. Hence, a vertical axis of symmetry passes through the midpoint of the horizontal side and through the bottom most vertex.

Label the regular pentagon in clockwise manner. Vertex 1 is the right end of the horizontal side. Vertex 2. Vertex 3 is the bottom most corner. Vertex 4. Vertex 5 is the left end of the horizontal side. Zero or O is the center of the pentagon and of the circle also. R is the radius of the circle.

So, O is (x,y) as you said.

Or,

xO = x

yO = y

Imagine, or draw a vertical-horizontal crossline at center O.

Imagine or draw radii R to each of the 5 vertices.

The pentagon is subdivided into 5 congruent isosceles triangles, whose apex angles are 360/5 = 72 degrees each.

In the isosceles triangle 1-O-5:

Angle O is bisected by the vertical axis of symmetry, so two congruent right triangles are formed, each of which has an angle of 73/2 = 36 degrees at point O.

So, the coordinates of vertex 1 are:

(x +R*sin(36deg), y +R*cos(36deg)) -------------answer.

Or,

X1 = x +Rsin(36deg)

y1 = y +Rcos(36deg)

In the isosceles triangle 1-O-2:

The horizontal crossline divides the angle O into 54deg and 18 deg.

{54deg is from 90 -36 = 54. 18deg is from 72 -54 = 18} ----****

So, the coordinates of Vertex 2 are:

(x +R*cos(18deg), y -R*sin(18deg)) ---------------answer.

Or,

X2 = x +Rcos(18deg)

y2 = y -Rsin(18deg)

The vertex 3 is on the vertical crossline.

So, the coordinates of Vertex 3 are:

(x,y -R) -------------------------------------answer.

Or,

x3 = x

y3 = y -R

See the explanation of the Vertex 2.

By symmetry, the coordinates of Vertex 4 are:

(x -R*cos(18deg), y -R*sin(18deg)) ---------------answer.

Or,

X4 = x -Rcos(18deg)

y4 = y -Rsin(18deg)

See explanation of Vertex 1.

By symmetery, the coordinates of Vertex 5 are:

(x -R*sin(36deg), y +R*cos(36deg)) -------------answer.

Or,

X5 = x -Rsin(36deg)

y5 = y +Rcos(36deg)