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Math Help - How to draw a pentagon

  1. #1
    Newbie
    Joined
    May 2006
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    6

    How to draw a pentagon

    Hi,

    I'm stuck up with this problem that I have.

    What I have is the center of the circle (x,y) and the radius.

    With these values is there any formula to find out the five co-ordinates which would form a regular pentagon.

    I need this to be implemented in a programming language so the formula needs to be something like

    x1=so and so
    y1= so and so

    x2 = soand so
    ..
    ..
    ..

    Can anyone please help me with this.

    I'm really stuck up and need a help.

    Thanks in advance.
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  2. #2
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    Quote Originally Posted by hari-kj
    Hi,

    I'm stuck up with this problem that I have.

    What I have is the center of the circle (x,y) and the radius.

    With these values is there any formula to find out the five co-ordinates which would form a regular pentagon.

    I need this to be implemented in a programming language so the formula needs to be something like

    x1=so and so
    y1= so and so

    x2 = soand so
    ..
    ..
    ..

    Can anyone please help me with this.

    I'm really stuck up and need a help.

    Thanks in advance.
    I do not know exactly what you mean, but here is one way.

    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)
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  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
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    Thanks
    692
    Hello, hari-kj!

    I have the center of the circle (x_1,y_1) and the radius r.

    Is there a formula to find the five coordinates which would form a regular pentagon?
    I worked this out a generation ago . . . in BASIC on my Tandy-1000 . . . LOL!

    x\:=\:x_1 + r\cdot\sin\left(\frac{2\pi}{5}n\right) . . . y \:=\:y_1 - r\cdot\cos\left(\frac{2\pi}{5}n\right) . . . and let n = 0,1,2,3,4


    The first point is at the 12:00 position; the points are plotted clockwise.
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