
How to construct a pentagon, a regular 5sided polygon, inscribed in a given circle?

1. 
 Start with the given circle.


2. 
 Use the
construction of the horizontal and vertical centerlines to create these lines.


3. 
 Draw an arc with its center at the endpoint of the horizontal centerline on the right, and passing through the intersection of both centerlines 2 (and therefore passing through the center of circle 1). The arc must intersect circle 1 twice.


4. 
 Draw the connecting line between both intersections of arc 3 with circle 1. This is the perpendicular bisector of the right half of the horizontal centerline, dividing that half into two equal parts.


5. 
 Draw the connecting line between the upper endpoint of the vertical centerline and the intersection of line 4 with the horizontal centerline.


6. 
 Draw an arc with its center at the last mentioned intersection (of line 4 with the horizontal centerline), passing through the center of circle 1. So, its radius is half that of circle 1. It must intersect line 5.


7. 
 Draw an arc with its center at the upper endpoint of the vertical centerline, passing through the intersection of arc 6 with line 5. It must intersect circle 1 twice.


8. 
 Draw the connecting line between both intersections of arc 7 with circle 1. This is one side of the pentagon.


9. 
 Draw an arc with its center at one endpoint of line 8, passing through the other endpoint. The arc must intersect circle 1 a second time.


10. 
 Draw the connecting line between the first endpoint of line 8 (step 9) and the second intersection of arc 9 with circle 1.


11. 
 Draw the connecting line between the second intersection of arc 9 with circle 1 and the lower endpoint of the vertical centerline.


12. 
 Repeat step 9 with the endpoints of line 8 exchanged, mirrored with respect to the vertical centerline.


13. 
 Repeat step 10 for the exchanged points, mirrored with respect to the vertical centerline.


14. 
 Repeat step 11 for the exchanged points, mirrored with respect to the vertical centerline.


15. 
 Lines 8, 10, 11, 13 and 14 together form the pentagon to be constructed.
This construction of a pentagon (there are different ones) can be used directly to construct a decagon, a regular 10sided polygon. Since line 8 (one side of the pentagon) is constructed by both intersections of arc 7 with circle 1, the center of this arc has equal distances to both intersections. The connections of this center with both intersections form two sides of the decagon.
The pentagon is characterised by the "golden section". This is the ratio of two numbers a and b (a < b), for which the following relation holds:
a / b = b / (a + b)
In words: the ratio of the smaller to the larger is equal to the ratio of the larger to the total. In numbers: a / b = 0.618..., b / a = 1 + (a / b) = 1.618... . Very interesting information about the golden section can be found in R. Knott's personal pages.
If the length of one side of the pentagon equals a, then the length of the diagonal equals b, or: the ratio of a side to a diagonal of a pentagon is equal to the golden section.

