N I N E
CONSTRUCTIONS FOR THE
REGULAR
P E N T A G O N
By Robin Hu
Most websites out there will give you only one or two different constructions for the regular pentagon. I have put together a collection of 9 different constructions for the regular pentagon. The algorithms range from very easy, to extremely complex, to a rare Mascheroni construction! The construction of the pentagon is elegant and it has been around since the ancient Greeks!
Inscribing a Pentagon in a given circle.
Step 1. Given circle O. Draw a diameter AB to circle O. Find the bisector E of segment OB. Find the perpendicular bisector of AB. Call it C.
Step 2. Construct a circle at point E, and with radius EC. Circle E intersects AB at D.
Step 3. Construct a circle at point C, and with radius CD. Circle C intersects circle O at P1. Line CP1 is a side of the pentagon. Use it to find the other 4 sides.
If you bisect one of the sides of the pentagon, you get a side of a Decagon (10 sides).
Lets say you want to construct a pentagon, yet you only know one side. Then you would follow this algorithm.
Given line AB, construct a pentagon with sides equal to AB.
Step 1. Draw a circle at A with radius AB. Draw a circle at B with radius AB. Call their intersections C and D.
Step 2. Connect line CD. Draw a circle at D with radius DA. Circle D intersects line CD at E. Circle D intersects circle A at F and intersects circle B at G.
Step 3. Draw a line through FE and GE. Line FE intersects circle B at H. Line GE intersects circle A at I.
Step 4. Draw an arc at I with radius IA. Draw an arc at H with radius HB. Arcs I and H intersect at J.
Step 5. All the points A,B,I,H,and J are points of the pentagon.
NEW!! See Regular Pentagon for details. |
1. First, draw a circle centered at point O. 2. Draw a line DB, the diameter of circle O. 3. Draw a circle at point A, with radius OA. 4. Draw a line through points A and B, intersecting the small circle at C. 5. Draw a circle centered at B, with radius BC. 6. Circle B intersects the original circle at points E and F. Points D, E and F are 3 points of the pentagon. Use them to find the other 2. |
1. Draw a circle centered at O. 2. Find the midpoint M of segment OC. 3. Connect points M and A. 4. Find MD, the bisector angle OMA. 5. From point D, draw a perpendicular line so it intersects the original circle at B. 6. AB is a side of the pentagon. | NEW!! See MathWorld for more details. |
NEW!! Click to Enlarge. See here for more details. |
1. Draw a line segment OA. 2. Find the midpoint M, of OA. 3. Draw a circle centered at M, with radius OA. 4. Draw a perpendicular line at M, intersecting circle M at points B and C. 5. Draw a circle centered at B with radius BO. Draw a line connecting B and O. 6. Draw a small circle at O, with radius OM. This circle intersects line BO at point D. 7. Draw a circle at point B with radius BC. 8. Draw a line perpendicular to OB through D. This line intersects the circle you just drew at points E and F. 9. Connect points EB and FB. The lines intersect circle B at points H and G. 10. Points O, H, and G are points of the pentagon. |
1. Draw a circle centered at O. 2. Mark a random point on circle O, call it A. Connect OA with a line. 3. Find the perpendicular bisector of OA. The midpoint of OA is B. 4. Find the midpoint of OB. Call it C. 5. Draw a circle at point B with radius BC. That circle intersects the line you just drew at point D. 6. Draw a ray through OD. The line intersects the original circle at point J. 7. Draw a circle at point D with radius DB. This circle intersects line OD at E. 8. Draw a perpendicular line through E. That line intersects the original circle at point K. 9. Points J and K are two points of the pentagon. |
NEW!!! See the University of Toronto for more details. |
NEW!! |
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Constructing a pentagon knowing only one side.
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NEW!! See the Golden Ratio for details. |
NEW!! See Mascheroni for details. |
A Mascheroni Construction is a constructions using compass only. Here we construct a pentagon drawing only circles and no lines. The construction is impractical for conventional use, but it's interesting to see how it's done anyway.
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Square | Triangle and Hexagon | Heptagon | Golden Ratio | Nonagon |
Pentadecagon | Heptadecagon | N-Gon | Squaring a Circle | Basic Constructions |