This page shows how to construct (draw) a regular hexagon inscribed in a circle with a compass and straightedge or ruler. This is the largest hexagon that will fit in the circle, with each vertex touching the circle. In a regular hexagon, the side length is equal to the distance from the center to a vertex, so we use this fact to set the compass to the proper side length, then step around the circle marking off the vertices.
Printable step-by-step instructions
The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.
Explanation of method
As can be seen in Definition of a Hexagon, each side of a regular hexagon is equal to the distance from the center to any vertex. This construction simply sets the compass width to that radius, and then steps that length off around the circle to create the six vertices of the hexagon.Proof
The image below is the final drawing from the above animation, but with the vertices labelled.
| Argument | Reason | |
|---|---|---|
| 1 | A,B,C,D,E,F all lie on the circle O | By construction. |
| 2 | AB = BC = CD = DE = EF | They were all drawn with the same compass width. |
| From (2) we see that five sides are equal in length, but the last side FA was not drawn with the compasses. It was the "left over" space as we stepped around the circle and stopped at F. So we have to prove it is congruent with the other five sides. | ||
| 3 | OAB is an equilateral triangle | AB was drawn with compass width set to OA, and OA = OB (both radii of the circle). |
| 4 | m∠AOB = 60° | All interior angles of an equilateral triangle are 60°. |
| 5 | m∠AOF = 60° | As in (4) m∠BOC, m∠COD, m∠DOE, m∠EOF are all &60deg; Since all the central angles add to 360°, m∠AOF = 360 - 5(60) |
| 6 | Triangle BOA, AOF are congruent | SAS See Test for congruence, side-angle-side. |
| 7 | AF = AB | CPCTC - Corresponding Parts of Congruent Triangles are Congruent |
| So now we have all the pieces to prove the construction | ||
| 8 | ABCDEF is a regular hexagon inscribed in the given circle |
|
- Q.E.D
Try it yourself
Click here for a printable worksheet containing two problems to try. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.Other constructions pages on this site
Lines
- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular at a point on a line
- Perpendicular from a line through a point
- Perpendicular from endpoint of a ray
- Divide a segment into n equal parts
- Parallel line through a point (angle copy)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)
Angles
- Bisecting an angle
- Copy an angle
- Construct a 30° angle
- Construct a 45° angle
- Construct a 60° angle
- Construct a 90° angle (right angle)
- Sum of n angles
- Difference of two angles
- Supplementary angle
- Complementary angle
- Constructing 75° 105° 120° 135° 150° angles and more
Triangles
- Copy a triangle
- Isosceles triangle, given base and side
- Isosceles triangle, given base and altitude
- Isosceles triangle, given leg and apex angle
- Equilateral triangle
- 30-60-90 triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and adjacent angles (asa)
- Triangle, given two angles and non-included side (aas)
- Triangle, given two sides and included angle (sas)
- Triangle medians
- Triangle midsegment
- Triangle altitude
- Triangle altitude (outside case)
Right triangles
- Right Triangle, given one leg and hypotenuse (HL)
- Right Triangle, given both legs (LL)
- Right Triangle, given hypotenuse and one angle (HA)
- Right Triangle, given one leg and one angle (LA)
Triangle Centers
Circles, Arcs and Ellipses
- Finding the center of a circle
- Circle given 3 points
- Tangent at a point on the circle
- Tangents through an external point
- Tangents to two circles (external)
- Tangents to two circles (internal)
- Incircle of a triangle
- Focus points of a given ellipse
- Circumcircle of a triangle
Polygons
- Square given one side
- Square inscribed in a circle
- Hexagon given one side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle
Non-Euclidean constructions
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