Polygons - Revision history

DT Online at 10:04, 7 June 2018

2018-06-07T10:04:12Z

← Older revision		Revision as of 10:04, 7 June 2018
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	\| [[File:CompassHex.png\|250px\|right]]		\| [[File:CompassHex.png\|250px\|right]]
	\|-		\|-
	\| '''~~Septagon~~ (aka ~~Heptagon~~)'''		\| '''Heptagon (aka Septagon)'''
	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* Draw a line of length equal to the length of side required and construct its perpendicular bisector - (see '''[[Basic Constructions]]''').		* Draw a line of length equal to the length of side required and construct its perpendicular bisector - (see '''[[Basic Constructions]]''').

DT Online at 08:14, 7 June 2018

2018-06-07T08:14:33Z

← Older revision		Revision as of 08:14, 7 June 2018
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	* Draw a line of length equal to the length of side required.		* Draw a line of length equal to the length of side required.
	* From one end of the line construct a perpendicular - (see '''[[Basic Constructions]]''').		* From one end of the line construct a perpendicular - (see '''[[Basic Constructions]]''').
	* Set the line length as a radius and strike an arc along the ~~perendicular~~ to find the second side of the square.		* Set the line length as a radius and strike an arc along the perpendicular to find the second side of the square.
	* Keep the same radius and strike arcs from the ends of the two adjacent sides as shown		* Keep the same radius and strike arcs from the ends of the two adjacent sides as shown
	* Joining to where the two arcs intersect provides the remaining two sides.		* Joining to where the two arcs intersect provides the remaining two sides.
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	----		----
	<span style="color: green">'''Note:'''		<span style="color: green">'''Note:'''
	Polygons with odd numbered sides have an apex at top centre and this can be used for greater accuracy as an ~~alternatve~~ starting point, or as check, when stepping off sides round the circle .		Polygons with odd numbered sides have an apex at top centre and this can be used for greater accuracy as an alternative starting point, or as check, when stepping off sides round the circle .
	</span>		</span>
	----		----
	<span style="color: green">'''Note:'''		<span style="color: green">'''Note:'''
	This construction may be used for other polygons by stepping off an appropriate number of sides around the semi-circle - but in all cases always start by drawing a ~~lne~~ through the 2nd division to establish the circumscribing circle centre.		This construction may be used for other polygons by stepping off an appropriate number of sides around the semi-circle - but in all cases always start by drawing a line through the 2nd division to establish the circumscribing circle centre.
	</span>		</span>
	----		----
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	----		----
	<span style="color: green">'''Note:'''		<span style="color: green">'''Note:'''
	Polygons with even numbers of sides have diagonals ~~parrallel~~ to the sides and this can be used as a check on accuracy		Polygons with even numbers of sides have diagonals parallel to the sides and this can be used as a check on accuracy
	</span>		</span>
	----		----
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	</span>		</span>
	----		----
	* Set radius to half the distance between c4 and c6 and step off above c6 to create c7 - the centre of a circle which circumscribes a ~~Septagon~~ ''(~~or Heptagon~~)''		* Set radius to half the distance between c4 and c6 and step off above c6 to create c7 - the centre of a circle which circumscribes a Heptagon ''(aka Septagon)''
	* Set radius to the length of side and step off round the circle.		* Set radius to the length of side and step off round the circle.
	</span>		</span>

DT Online: Added Template

2016-06-02T19:13:41Z

Added Template

← Older revision		Revision as of 19:13, 2 June 2016
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			__TOC__


			=====Description=====

	Drawing regular [https://en.wikipedia.org/wiki/Polygon '''Polygons'''] is now most easily accomplished using modern [https://en.wikipedia.org/wiki/Technical_drawing '''drafting'''] aids and [https://en.wikipedia.org/wiki/Computer-aided_design '''Computer Aided Design'''] tools. But there are occasions when it is useful to have an understanding of some basic constructions using just rule and compasses ''(or pegs and pieces of string)''. Examples might include large scale work for stage sets or carnival floats, or on-site work for buildings, playgrounds, sports fields and gardens.		Drawing regular [https://en.wikipedia.org/wiki/Polygon '''Polygons'''] is now most easily accomplished using modern [https://en.wikipedia.org/wiki/Technical_drawing '''drafting'''] aids and [https://en.wikipedia.org/wiki/Computer-aided_design '''Computer Aided Design'''] tools. But there are occasions when it is useful to have an understanding of some basic constructions using just rule and compasses ''(or pegs and pieces of string)''. Examples might include large scale work for stage sets or carnival floats, or on-site work for buildings, playgrounds, sports fields and gardens.

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	</span>		</span>
	----		----

			=====Constructions=====

	{\| cellpadding="5"		{\| cellpadding="5"
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	\| [[File:CompassOctagon.png\|250px\|right]]		\| [[File:CompassOctagon.png\|250px\|right]]
	\|}		\|}

			{{Drawing Instruments Buyers Guide}}


	[[Category:Secondary]]		[[Category:Secondary]]
	[[Category:Geometry]]		[[Category:Geometry]]

DT Online at 12:06, 29 July 2015

2015-07-29T12:06:18Z

← Older revision		Revision as of 12:06, 29 July 2015
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	\| [[File:CompassHex.png\|250px\|right]]		\| [[File:CompassHex.png\|250px\|right]]
	\|-		\|-
	\| '''Septagon'''		\| '''Septagon (aka Heptagon)'''
	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* Draw a line of length equal to the length of side required and construct its perpendicular bisector - (see '''[[Basic Constructions]]''').		* Draw a line of length equal to the length of side required and construct its perpendicular bisector - (see '''[[Basic Constructions]]''').
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	----		----
	<span style="color: green">'''Note:'''		<span style="color: green">'''Note:'''
	~~This~~ circle would circumscribe a pentagon and provides an alternative construction.		A circle centred here would circumscribe a pentagon and provides an alternative construction.
	</span>		</span>
	----		----

DT Online at 12:02, 29 July 2015

2015-07-29T12:02:31Z

← Older revision		Revision as of 12:02, 29 July 2015
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	* Draw a line of length equal to the length of side required and construct its perpendicular bisector - (see '''[[Basic Constructions]]''').		* Draw a line of length equal to the length of side required and construct its perpendicular bisector - (see '''[[Basic Constructions]]''').
	* Draw a semi-circle at one end of the line and divide its circumference into as many equal parts as there are polygon sides required ''(5 in this case)''		* Draw a semi-circle at one end of the line and divide its circumference into as many equal parts as there are polygon sides required ''(5 in this case)''
	* The line connecting the semi-circle centre to the 2nd point is a side of the required polygon ''(pentagon) in this case.		* The line connecting the semi-circle centre to the 2nd point is a side of the required polygon ''(pentagon in this case)''.
	* Construct the perpendicular bisector of this second side to intersect with that drawn for the base line.		* Construct the perpendicular bisector of this second side to intersect with that drawn for the base line.
	* Where the two bisectors intersect is the centre for a [https://en.wikipedia.org/wiki/Circumscribed_circle '''circumscribing circle'''] to the required polygon.		* Where the two bisectors intersect is the centre for a [https://en.wikipedia.org/wiki/Circumscribed_circle '''circumscribing circle'''] to the required polygon.

DT Online: Completed Pentagon

2015-02-25T18:52:37Z

Completed Pentagon

← Older revision		Revision as of 18:52, 25 February 2015
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	* Draw a line of length equal to the length of side required.		* Draw a line of length equal to the length of side required.
	* From each end of the line, strike arcs using the line length as radius.		* From each end of the line, strike arcs using the line length as radius.
	* Where they intersect will provide the top apex of the required triangle. </span>		* Where they intersect will provide the top [https://en.wikipedia.org/wiki/Apex_%28geometry%29 '''apex'''] of the required triangle. </span>
	\| [[File:CompassTriangle.png\|250px\|right]]		\| [[File:CompassTriangle.png\|250px\|right]]
	\|-		\|-
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	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* Draw a line of length equal to the length of side required and construct its perpendicular bisector - (see '''[[Basic Constructions]]''').		* Draw a line of length equal to the length of side required and construct its perpendicular bisector - (see '''[[Basic Constructions]]''').
			* Draw a semi-circle at one end of the line and divide its circumference into as many equal parts as there are polygon sides required ''(5 in this case)''
			* The line connecting the semi-circle centre to the 2nd point is a side of the required polygon ''(pentagon) in this case.
			* Construct the perpendicular bisector of this second side to intersect with that drawn for the base line.
			* Where the two bisectors intersect is the centre for a [https://en.wikipedia.org/wiki/Circumscribed_circle '''circumscribing circle'''] to the required polygon.
			* Draw the circle and step off the required number of sides.
	----		----
	<span style="color: green">'''Note:'''		<span style="color: green">'''Note:'''
	Polygons with odd numbered sides have an apex at top centre and this can be used for greater accuracy as an alternatve starting point, or as check, when stepping off sides round the circle .		Polygons with odd numbered sides have an apex at top centre and this can be used for greater accuracy as an alternatve starting point, or as check, when stepping off sides round the circle .
			</span>
			----
			<span style="color: green">'''Note:'''
			This construction may be used for other polygons by stepping off an appropriate number of sides around the semi-circle - but in all cases always start by drawing a lne through the 2nd division to establish the circumscribing circle centre.
	</span>		</span>
	----		----

DT Online: Completed text and added category links

2015-02-25T18:36:24Z

Completed text and added category links

← Older revision		Revision as of 18:36, 25 February 2015
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	* Bisect the exterior angles at the base to create 45<sup>0</sup> angles as shown.		* Bisect the exterior angles at the base to create 45<sup>0</sup> angles as shown.
	* Where these bisectors cut the semi-circles are two sides of the octagon.		* Where these bisectors cut the semi-circles are two sides of the octagon.
	* Draw perpendiculars from where the 45<sup>0</sup> lines intersect the semi-circles ~~and scribe~~ two ~~more~~ lengths of ~~side along them~~</span>		* Draw lines from the other end of the line through the top centre of the semi-circle - these will be parallel to the bisectors.
			* Draw perpendiculars from where the 45<sup>0</sup> lines intersect the semi-circles up to their intersection with lines through the tops of the semi-circles - these are two vertical sides of the octagon.
			* Complete the octagon by scribing lengths of sides from the ends of these vertical sides on to the first perpendiculars drawn as shown.
			* Check the accuracy of all points by ensuring diagonals are parallel to sides.
			</span>
	\| [[File:CompassOctagon.png\|250px\|right]]		\| [[File:CompassOctagon.png\|250px\|right]]
	\|}		\|}


			[[Category:Secondary]]
			[[Category:Geometry]]

DT Online at 17:23, 25 February 2015

2015-02-25T17:23:29Z

← Older revision		Revision as of 17:23, 25 February 2015
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	----		----
	<span style="color: green">'''Note:'''		<span style="color: green">'''Note:'''
	~~###~~		Polygons with even numbers of sides have diagonals parrallel to the sides and this can be used as a check on accuracy
	</span>		</span>
	----		----
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	\| '''Octagon'''		\| '''Octagon'''
	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* ~~###~~ </span>		* Draw a line of length equal to the length of side required and draw a semi-circle from each end
			* Use the semi-circle arcs to construct perpendiculars from each end of the line - (see '''[[Basic Constructions]]''').
			* Bisect the exterior angles at the base to create 45<sup>0</sup> angles as shown.
			* Where these bisectors cut the semi-circles are two sides of the octagon.
			* Draw perpendiculars from where the 45<sup>0</sup> lines intersect the semi-circles and scribe two more lengths of side along them</span>
	\| [[File:CompassOctagon.png\|250px\|right]]		\| [[File:CompassOctagon.png\|250px\|right]]
	\|}		\|}

DT Online: Added septagon

2015-02-25T17:04:44Z

Added septagon

← Older revision		Revision as of 17:04, 25 February 2015
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	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* Draw a line of length equal to the length of side required and construct its perpendicular bisector - (see '''[[Basic Constructions]]''').		* Draw a line of length equal to the length of side required and construct its perpendicular bisector - (see '''[[Basic Constructions]]''').
	* From one end of the base line construct 45<~~sup~~>~~0</sup> and 60<sup>0</sup> angles.~~		----
	* Where these to angles cross the vertical centre ~~line would mark~~ the ~~centres of circumscribing circles for squares and hexagons respectively~~		<span style="color: green">'''Note:'''
	* Bisect between these two centres and draw a circle ~~which circumscribes the pentagon~~.		Polygons with odd numbered sides have an apex at top centre and this can be used for greater accuracy as an alternatve starting point, or as check, when stepping off sides round the circle .
	* Set radius to the length of side		</span>
			----
	</span>		</span>
	\| [[File:~~CompassPentagon~~.png\|250px\|right]]		\| [[File:CompassPentagon2.png\|250px\|right]]
	\|-		\|-
	\| '''Hexagon'''		\| '''Hexagon'''
	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* ### </span>		* Draw a line of length equal to the length of side required and construct an equilateral triangle on this as the base ''(see above)''
			* Set radius to length of side and draw the circumscribing circle for the hexagon from the top apex of the triangle.
			* With same radius set, step off sides around the circle as shown.
			----
			<span style="color: green">'''Note:'''
			###
			</span>
			----
			</span>
	\| [[File:CompassHex.png\|250px\|right]]		\| [[File:CompassHex.png\|250px\|right]]
			\|-
			\| '''Septagon'''
			\| <span style="color:#B00000">
			* Draw a line of length equal to the length of side required and construct its perpendicular bisector - (see '''[[Basic Constructions]]''').
			* From one end of the base line construct 45<sup>0</sup> and 60<sup>0</sup> angles.
			* Where these to angles cross the vertical centre line would mark the centres of circumscribing circles for squares and hexagons respectively
			* Bisect between these two centres.
			----
			<span style="color: green">'''Note:'''
			This circle would circumscribe a pentagon and provides an alternative construction.
			</span>
			----
			* Set radius to half the distance between c4 and c6 and step off above c6 to create c7 - the centre of a circle which circumscribes a Septagon ''(or Heptagon)''
			* Set radius to the length of side and step off round the circle.
			</span>
			\| [[File:CompassSeptagon.png\|250px\|right]]
	\|-		\|-
	\| '''Octagon'''		\| '''Octagon'''

DT Online: Creat Polygons

2015-02-25T12:48:28Z

Creat Polygons

New page

Drawing regular [https://en.wikipedia.org/wiki/Polygon '''Polygons'''] is now most easily accomplished using modern [https://en.wikipedia.org/wiki/Technical_drawing '''drafting'''] aids and [https://en.wikipedia.org/wiki/Computer-aided_design '''Computer Aided Design'''] tools. But there are occasions when it is useful to have an understanding of some basic constructions using just rule and compasses ''(or pegs and pieces of string)''. Examples might include large scale work for stage sets or carnival floats, or on-site work for buildings, playgrounds, sports fields and gardens.

The following examples are limited to constructions of [http://www.mathopenref.com/polygonregular.html '''regular polygons''']. [http://www.mathopenref.com/polygonirregular.html '''Irregular polygon ''']construction may be achieved using a process of [https://en.wikipedia.org/wiki/Triangulation triangulation], [https://en.wikipedia.org/wiki/Euclidean_vector vectors], [https://en.wikipedia.org/wiki/Coordinate_system coordinates] or plotting points on a [https://en.wikipedia.org/wiki/Matrix_%28mathematics%29 matrix] for example. Some other polygons ''(e.g. [http://www.mathopenref.com/quadrilateralinscribed.html Cyclic Quadlitateral])'' have particular properties which can be helpful to know when constructing them.

See also the YouTube video [http://youtu.be/LBgIWQcC6lM '''All the possible polygons!''']. This shows how all regular polygons can be constructed using classical geometry techniques
----
'''Note:'''
Some constructions on the YouTube video may be different to those described below - there are several methods of construction possible for most regular polygons.

----

{| cellpadding="5"
|-
| '''Equilateral Triangle'''
| 
* Draw a line of length equal to the length of side required.
* From each end of the line, strike arcs using the line length as radius.
* Where they intersect will provide the top apex of the required triangle. 
| [[File:CompassTriangle.png|250px|right]]
|-
| '''Square'''
| 
* Draw a line of length equal to the length of side required.
* From one end of the line construct a perpendicular - (see '''[[Basic Constructions]]''').
* Set the line length as a radius and strike an arc along the perendicular to find the second side of the square.
* Keep the same radius and strike arcs from the ends of the two adjacent sides as shown
* Joining to where the two arcs intersect provides the remaining two sides.

| [[File:CompassSquare.png|250px|right]]
|-
| '''Pentagon'''
| 
* Draw a line of length equal to the length of side required and construct its perpendicular bisector - (see '''[[Basic Constructions]]''').
* From one end of the base line construct 450 and 600 angles.
* Where these to angles cross the vertical centre line would mark the centres of circumscribing circles for squares and hexagons respectively
* Bisect between these two centres and draw a circle which circumscribes the pentagon.
* Set radius to the length of side

| [[File:CompassPentagon.png|250px|right]]
|-
| '''Hexagon'''
| 
* ### 
| [[File:CompassHex.png|250px|right]]
|-
| '''Octagon'''
| 
* ### 
| [[File:CompassOctagon.png|250px|right]]
|}