Drawing a Hexagon Outside a Circle
This folio 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 middle to a vertex, so nosotros utilize this fact to set the compass to the proper side length, then step effectually the circle marking off the vertices.
Printable step-by-step instructions
The above blitheness is bachelor as a printable step-by-stride instruction sail, which can be used for making handouts or when a computer is not available.
Caption of method
As can exist 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 and then steps that length off around the circle to create the six vertices of the hexagon.
Proof
The image below is the concluding drawing from the in a higher place animation, but with the vertices labelled.
| Argument | Reason | |
|---|---|---|
| ane | A,B,C,D,E,F all prevarication on the circle O | By construction. |
| 2 | AB = BC = CD = DE = EF | They were all drawn with the same compass width. |
| From (2) nosotros see that five sides are equal in length, but the last side FA was not drawn with the compasses. Information technology was the "left over" space as nosotros stepped effectually the circle and stopped at F. So we have to prove it is coinciding 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 circumvolve). |
| 4 | m∠AOB = 60° | All interior angles of an equilateral triangle are 60°. |
| 5 | m∠AOF = lx° | Every bit in (4) m∠BOC, m∠COD, chiliad∠DOE, one thousand∠EOF are all &60deg; Since all the central angles add to 360°, m∠AOF = 360 - 5(sixty) |
| vi | Triangle BOA, AOF are congruent | SAS See Test for congruence, side-bending-side. |
| vii | AF = AB | CPCTC - Corresponding Parts of Congruent Triangles are Coinciding |
| And then now we have all the pieces to evidence the construction | ||
| 8 | ABCDEF is a regular hexagon inscribed in the given circle |
|
- Q.Eastward.D
Attempt information technology yourself
Click here for a printable worksheet containing ii issues to effort. When y'all become to the folio, use the browser print command to print as many every bit you wish. The printed output is non copyright.
Other constructions pages on this site
- List of printable constructions worksheets
Lines
- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Deviation of two line segments
- Perpendicular bisector of a line segment
- Perpendicular at a point on a line
- Perpendicular from a line through a indicate
- Perpendicular from endpoint of a ray
- Split up a segment into n equal parts
- Parallel line through a point (bending copy)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)
Angles
- Bisecting an angle
- Copy an angle
- Construct a thirty° angle
- Construct a 45° bending
- Construct a 60° angle
- Construct a ninety° angle (right angle)
- Sum of n angles
- Difference of ii 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 distance
- Isosceles triangle, given leg and noon bending
- Equilateral triangle
- 30-60-ninety triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and next 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
- Correct Triangle, given 1 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 bending (LA)
Triangle Centers
- Triangle incenter
- Triangle circumcenter
- Triangle orthocenter
- Triangle centroid
Circles, Arcs and Ellipses
- Finding the center of a circle
- Circumvolve given three points
- Tangent at a point on the circumvolve
- 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
- Foursquare given one side
- Square inscribed in a circumvolve
- Hexagon given 1 side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle
Non-Euclidean constructions
- Construct an ellipse with string and pins
- Find the center of a circle with any correct-angled object
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