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!

Pentagon Constructing the Pentagon and Decagon.

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).


Constructing a pentagon, knowing only one side.

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!!
A THIRD CONSTRUCTION!
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!!
Another Neat Construction!
See MathWorld for more details.
NEW!!
A very complex construction!!
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!!!
A SIXTH CONSTRUCTION!!! THIS IS PRETTY EASY!
See the University of Toronto for more details.
NEW!!
Constructing a pentagon knowing a diagnol.
  1. Draw a line segment AB (yellow).
  2. Draw a circle at A and B, both with radius AB (yellow).
  3. Find the midpoint of AB, C. Draw a perpendicular line through C.
  4. Draw a circle at C with radius AB (purple). This circle intersects the line you just drew at point D.
  5. Find E, the midpoint of DC. Connect points DB with a line.
  6. Draw a circle at point D with radius DE (blue). This circle intersects DB at F.
  7. Draw a circle at point B with radius BF (red). This circle intersects line DC at G. It also intersects the large circle A at point I.
  8. Draw a circle at point A with radius AG (red). This circle intersects the large circle B at point H.
  9. AGBIH is a pentagon.
Constructing a pentagon knowing only one side.
  1. Given side AB (yellow), find M, the midpoint of AB.
  2. Extend line AB and draw a perpendicular line through A.
  3. Draw a circle at A with radius AB (purple). This circle intersects the perpendicular line at C.
  4. At point M, draw a circle with radius MC (yellow). This circle intersects line AB at point D.
  5. Draw another circle at A, this time with radius AD (red). Draw a circle at B with radius AD (red) also. These two circle intersect at E.
  6. Draw a circle at point B with radius BA (purple). It intersects the large circle A at point G. Draw a circle at point A with radius AB. This circle intersects the large circle B at point F.
  7. ABGEF are points of a regular pentagon.
NEW!!
Constructing a pentagon knowing a side.
See the Golden Ratio for details.
NEW!!
A rare Mascheroni Construction!
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.
  1. Draw a circle centered at A (yellow). Mark a random point B on circle A. This is where the pentagon will be inscribed.
  2. Draw a circle at point B with radius AB (green). It intersects the original circle at C. Draw a circle at C with radius CA. It intersects at D. Draw a circle at D with radius DA. It intersects the original circle at E.
  3. Draw a circle at B with radius BD (cyan). Draw a circle at E of the same radius. The two large circles intersect at F.
  4. Draw a circle at point A with radius AF (purple). It intersects circle B (green) at G.
  5. At point G, draw a circle with radius GB (purple). It intersects the original circle at H.
  6. Draw a circle at H with radius HG (blue). It intersects the original circle at points I and J.
  7. At points I and J, draw a circle with radii ID and JC (red). The two circles intersect at K.
  8. Draw a circle at B with radius BK. This circle intersects the original circle at M and L.
  9. At point L draw a circle with radius LB (not shown). It intersects the original circle at N. Draw a circle at M with radius MB (not shown). It intersects the original circle at O.
  10. BMONL is a regular pentagon.

More Pages
Square Triangle and Hexagon Heptagon Golden Ratio Nonagon
Pentadecagon Heptadecagon N-Gon Squaring a Circle Basic Constructions

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© 2002 Robin Hu

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