How to Draw a Circle Inside a Hexagon

This folio shows how to construct (depict) a regular hexagon inscribed in a circle with a compass and straightedge or ruler. This is the largest hexagon that will fit in the circumvolve, with each vertex touching the circle. In a regular hexagon, the side length is equal to the distance from the center to a vertex, and so nosotros utilise this fact to set the compass to the proper side length, then stride around the circle marking off the vertices.

Printable step-by-pace instructions

The above animation is available every bit a printable step-by-step educational activity canvass, which can be used for making handouts or when a computer is not bachelor.

Explanation of method

Every bit 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 only sets the compass width to that radius, and then steps that length off around the circumvolve to create the six vertices of the hexagon.

Proof

The image below is the last drawing from the above 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 fatigued with the same compass width.
From (2) we see that five sides are equal in length, merely the terminal side FA was not fatigued with the compasses. Information technology was the "left over" space as we stepped around the circle and stopped at F. So we have to bear witness it is congruent with the other v sides.
iii	OAB is an equilateral triangle	AB was drawn with compass width set to OA, and OA = OB (both radii of the circumvolve).
4	yard∠AOB = lx°	All interior angles of an equilateral triangle are sixty°.
five	chiliad∠AOF = sixty°	Equally in (four) m∠BOC, m∠COD, one thousand∠DOE, thousand∠EOF are all &60deg; Since all the central angles add to 360°, yard∠AOF = 360 - 5(60)
6	Triangle BOA, AOF are congruent	SAS Encounter Examination for congruence, side-angle-side.
seven	AF = AB	CPCTC - Corresponding Parts of Congruent Triangles are Congruent
So at present we accept all the pieces to prove the structure
eight	ABCDEF is a regular hexagon inscribed in the given circle	From (1), all vertices lie on the circle From (20), (7), all sides are the same length The polygon has vi sides.

- Q.E.D

Endeavour information technology yourself

Click hither for a printable worksheet containing two problems to endeavor. When you get to the page, utilize the browser print control to print every bit many every bit you wish. The printed output is not copyright.

Other constructions pages on this site

List of printable constructions worksheets

Lines

Introduction to constructions
Re-create a line segment
Sum of north line segments
Divergence of two line segments
Perpendicular bisector of a line segment
Perpendicular at a point on a line
Perpendicular from a line through a betoken
Perpendicular from endpoint of a ray
Divide a segment into due north 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
Re-create an angle
Construct a 30° angle
Construct a 45° angle
Construct a 60° angle
Construct a 90° angle (right bending)
Sum of n angles
Difference of two angles
Supplementary bending
Complementary angle
Constructing 75° 105° 120° 135° 150° angles and more

Triangles

Re-create a triangle
Isosceles triangle, given base of operations and side
Isosceles triangle, given base and altitude
Isosceles triangle, given leg and apex angle
Equilateral triangle
30-threescore-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 bending (sas)
Triangle medians
Triangle midsegment
Triangle altitude
Triangle altitude (outside case)

Right triangles

Correct Triangle, given one leg and hypotenuse (HL)
Right Triangle, given both legs (LL)
Right Triangle, given hypotenuse and one angle (HA)
Correct Triangle, given one leg and one angle (LA)

Triangle Centers

Triangle incenter
Triangle circumcenter
Triangle orthocenter
Triangle centroid

Circles, Arcs and Ellipses

Finding the eye of a circle
Circle given three points
Tangent at a bespeak on the circumvolve
Tangents through an external signal
Tangents to two circles (external)
Tangents to 2 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 one side
Hexagon inscribed in a given circle
Pentagon inscribed in a given circle

Not-Euclidean constructions

Construct an ellipse with string and pins
Find the center of a circle with any right-angled object

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Source: https://www.mathopenref.com/constinhexagon.html