Circles - Revision history

DT Online: Added Template

2016-06-02T18:53:10Z

Added Template

← Older revision		Revision as of 18:53, 2 June 2016
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	[[File:CircleParts.png\|~~300px~~\|right]]		[[File:CircleParts.png\|400px\|right]]
			__TOC__


			=====Description=====

	[https://en.wikipedia.org/wiki/Circle '''Circles'''] are very familiar but have associated with them terminology, properties and characteristics with which all involved with '''[[Design and Technology\|design and technology]]''' should understand.		[https://en.wikipedia.org/wiki/Circle '''Circles'''] are very familiar but have associated with them terminology, properties and characteristics with which all involved with '''[[Design and Technology\|design and technology]]''' should understand.


	A particular property of a circle is that its '''radius''' can be stepped off around its '''circumference''' exactly 6 times. In effect, this constructs 6 equi-lateral triangles, the interior angles of which are all 60<sup>0</sup>, and the sum of all angles at the centre of <u>any</u> '''[[Polygons\|polygon]]''' must equal 360<sup>0</sup>.		A particular property of a circle is that its '''radius''' can be stepped off around its '''circumference''' exactly 6 times. In effect, this constructs 6 equi-lateral triangles, the interior angles of which are all 60<sup>0</sup>, and the sum of all angles at the centre of <u>any</u> '''[[Polygons\|polygon]]''' must equal 360<sup>0</sup>.


	The sides of these equi-lateral triangles become '''chords''' of the circle and should not be confused with the length of an '''arc''', which is longer. Lengths of arcs can be found by dividing the circumference by the number of sides. The circumference is calculated as 2πr or πD ''([https://en.wikipedia.org/wiki/Pi '''π'''] is a constant obtained by dividing circumference by diameter - i.e. approximately 3.142)".		The sides of these equi-lateral triangles become '''chords''' of the circle and should not be confused with the length of an '''arc''', which is longer. Lengths of arcs can be found by dividing the circumference by the number of sides. The circumference is calculated as 2πr or πD ''([https://en.wikipedia.org/wiki/Pi '''π'''] is a constant obtained by dividing circumference by diameter - i.e. approximately 3.142)".


			=====Constructions=====
	{\| cellpadding="5"		{\| cellpadding="5"
	\|-		\|-
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	\|}		\|}


			{{Drawing Instruments Buyers Guide}}


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

DT Online at 17:11, 13 August 2015

2015-08-13T17:11:19Z

← Older revision		Revision as of 17:11, 13 August 2015
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	\| '''Intersecting Chords Theorem'''		\| '''Intersecting Chords Theorem'''
	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* Looking at one of the chords and its bisector above, '''[[Intersecting Chords\|it can be proved''']] that '''A x B = C x D'''.<div style="clear: both;"></div>		* Looking at one of the chords and its bisector above, '''[[Intersecting Chords\|it can be proved]]''' that '''A x B = C x D'''.<div style="clear: both;"></div>
	* It can be derived from this that '''[[Intersecting Chords\|finding the centre of an arch]]''' doorway or window for example can be found by the formula:		* It can be derived from this that '''[[Intersecting Chords\|finding the centre of an arch]]''' doorway or window for example can be found by the formula:
	[[File:ArcRadius.png\|125px\|right]]		[[File:ArcRadius.png\|125px\|right]]

DT Online at 17:08, 13 August 2015

2015-08-13T17:08:21Z

← Older revision		Revision as of 17:08, 13 August 2015
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	\| '''Angle in a Semi-Circle'''		\| '''Angle in a Semi-Circle'''
	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* <u>~~All~~</u> ~~angles~~ drawn by connecting each end of a semi-circle diameter to point on its circumference is a right angle.		* <u>Any</u> angle drawn by connecting each end of a semi-circle diameter to point on its circumference is a right angle.
	----		----
	<span style="color: green">'''Note:'''		<span style="color: green">'''Note:'''

DT Online at 17:07, 13 August 2015

2015-08-13T17:07:18Z

← Older revision		Revision as of 17:07, 13 August 2015
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	\| '''Intersecting Chords Theorem'''		\| '''Intersecting Chords Theorem'''
	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* Looking at one of the chords and its bisector above, '''[[Intersecting Chords\|it can be proved''']] that '''A x B = C x D'''.		* Looking at one of the chords and its bisector above, '''[[Intersecting Chords\|it can be proved''']] that '''A x B = C x D'''.<div style="clear: both;"></div>
	* It can be derived from this that '''[[Intersecting Chords\|finding the centre of an arch]]''' doorway or window for example can be found by the formula:		* It can be derived from this that '''[[Intersecting Chords\|finding the centre of an arch]]''' doorway or window for example can be found by the formula:
	[[File:ArcRadius.png\|125px\|right]]		[[File:ArcRadius.png\|125px\|right]]

DT Online at 08:52, 27 February 2015

2015-02-27T08:52:09Z

← Older revision		Revision as of 08:52, 27 February 2015
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	\| '''Angle in a Semi-Circle'''		\| '''Angle in a Semi-Circle'''
	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* <u>~~Any~~</u> ~~angle~~ drawn ~~within~~ a semi-circle is a right angle.		* <u>All</u> angles drawn by connecting each end of a semi-circle diameter to point on its circumference is a right angle.
	----		----
	<span style="color: green">'''Note:'''		<span style="color: green">'''Note:'''

DT Online at 07:59, 27 February 2015

2015-02-27T07:59:31Z

← Older revision		Revision as of 07:59, 27 February 2015
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	\| '''Intersecting Chords Theorem'''		\| '''Intersecting Chords Theorem'''
	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* Looking at one of the chords and its bisector above, ~~[http://www.mathopenref.com/chordsintersecting.html~~ '''it can be proved'''] that '''A x B = C x D'''.		* Looking at one of the chords and its bisector above, '''[[Intersecting Chords\|it can be proved''']] that '''A x B = C x D'''.
	* It can be derived from this that ~~[http://www.mathopenref.com/arcradius.html~~ '''finding the centre of an arch'''] doorway or window for example can be found by the formula:		* It can be derived from this that '''[[Intersecting Chords\|finding the centre of an arch]]''' doorway or window for example can be found by the formula:
	[[File:ArcRadius.png\|125px\|right]]		[[File:ArcRadius.png\|125px\|right]]
	: Where: '''W''' is the length of the chord defining the base of the arc and '''H''' is the height measured at the midpoint of the arc's base.		: Where: '''W''' is the length of the chord defining the base of the arc and '''H''' is the height measured at the midpoint of the arc's base.
	----		----
	<span style="color: green">'''Note:'''		<span style="color: green">'''Note:'''
	This is a particular case of the '''[[Intersecting Chords Theorem]]''' which states that the relationship ''(A x B = C x D)'' is true for <u> any</u> two intersecting chords, whether or not one of them is a diameter		This is a particular case of the '''[[Intersecting Chords\|Intersecting Chords Theorem]]''' which states that the relationship ''(A x B = C x D)'' is true for <u> any</u> two intersecting chords, whether or not one of them is a diameter
	</span>		</span>
	----		----

DT Online at 00:05, 26 February 2015

2015-02-26T00:05:40Z

← Older revision		Revision as of 00:05, 26 February 2015
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	[https://en.wikipedia.org/wiki/Circle '''Circles'''] are very familiar but have associated with them terminology, properties and characteristics with which all involved with '''[[Design and Technology\|design and technology]]''' should understand.		[https://en.wikipedia.org/wiki/Circle '''Circles'''] are very familiar but have associated with them terminology, properties and characteristics with which all involved with '''[[Design and Technology\|design and technology]]''' should understand.

	A particular property of a circle is that its '''radius''' can be stepped off around its '''circumference''' exactly 6 times. ~~This is effectively constructing~~ 6 equi-lateral triangles, the interior angles of which are all 60<sup>0</sup>, and the sum of all angles at the centre of <u>any</u> '''[[Polygons\|polygon]]''' must equal 360<sup>0</sup>.		A particular property of a circle is that its '''radius''' can be stepped off around its '''circumference''' exactly 6 times. In effect, this constructs 6 equi-lateral triangles, the interior angles of which are all 60<sup>0</sup>, and the sum of all angles at the centre of <u>any</u> '''[[Polygons\|polygon]]''' must equal 360<sup>0</sup>.

	The sides of these equi-lateral triangles become '''chords''' of the circle and should not be confused with the length of an '''arc''', which is longer. Lengths of arcs can be found by dividing the circumference by the number of sides. The circumference is calculated as 2πr or πD ''([https://en.wikipedia.org/wiki/Pi '''π'''] is a constant obtained by dividing circumference by diameter - i.e. approximately 3.142)".		The sides of these equi-lateral triangles become '''chords''' of the circle and should not be confused with the length of an '''arc''', which is longer. Lengths of arcs can be found by dividing the circumference by the number of sides. The circumference is calculated as 2πr or πD ''([https://en.wikipedia.org/wiki/Pi '''π'''] is a constant obtained by dividing circumference by diameter - i.e. approximately 3.142)".
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	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* Looking at one of the chords and its bisector above, [http://www.mathopenref.com/chordsintersecting.html '''it can be proved'''] that '''A x B = C x D'''.		* Looking at one of the chords and its bisector above, [http://www.mathopenref.com/chordsintersecting.html '''it can be proved'''] that '''A x B = C x D'''.
	* It can be derived from this that [http://www.mathopenref.com/arcradius.html '''finding the centre of an arch'''] doorway or window for example can be found by the formula: [[File:ArcRadius.png\|125px]]		* It can be derived from this that [http://www.mathopenref.com/arcradius.html '''finding the centre of an arch'''] doorway or window for example can be found by the formula:
			[[File:ArcRadius.png\|125px\|right]]
	: Where: '''W''' is the length of the chord defining the base of the arc and '''H''' is the height measured at the midpoint of the arc's base.		: Where: '''W''' is the length of the chord defining the base of the arc and '''H''' is the height measured at the midpoint of the arc's base.
	* This is a particular case of the '''[[Intersecting Chords Theorem]]''' which states that the relationship ''(A x B = C x D)'' is true for <u> any</u> two intersecting chords, whether or not one of them is a diameter</span>		----
			<span style="color: green">'''Note:'''
			This is a particular case of the '''[[Intersecting Chords Theorem]]''' which states that the relationship ''(A x B = C x D)'' is true for <u> any</u> two intersecting chords, whether or not one of them is a diameter
			</span>
			----
			</span>
	\| [[File:ChordsTheorem.png\|250px\|right]]		\| [[File:ChordsTheorem.png\|250px\|right]]
	\|-		\|-
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	\| <span style="color:#B00000">		\| <span style="color:#B00000">
	* For most practical purposes, especially with large-scale work, it can be satisfactory simply to draw a line touching a circle as the '''tangent'''		* For most practical purposes, especially with large-scale work, it can be satisfactory simply to draw a line touching a circle as the '''tangent'''
	* There may be occasions however when it is important to know exactly where the tangent touches the circle circumference ''(i.e. the Point of Tangency]]''		* There may be occasions however when it is important to know exactly where the tangent touches the circle circumference ''(i.e. the Point of Tangency)''
	* For points on the circumference, the tangent is simply drawn as a line at right angles to a line joining the point to the circle centre ''(i.e. the Normal)''.		* For points on the circumference, the tangent is simply drawn as a line at right angles to a line joining the point to the circle centre ''(i.e. the Normal)''.
	* Points outside the circumference exploit the property of ''Angles in a Semi-Circle'' as described above.		* Points outside the circumference exploit the property of ''Angles in a Semi-Circle'' as described above.
	* ~~Join~~ the point to the circle centre and construct a semi-circle on this line.		* For points outside the circle, join the point to the circle centre and construct a semi-circle on this line.
	* Where the semi-circle cuts the circumference provides the ''Point of Tangency'' to which may be drawn both the ''Normal'' and ''Tangent'' required.		* Where the semi-circle cuts the circumference provides the ''Point of Tangency'' to which may be drawn both the ''Normal'' and ''Tangent'' required.
	----		----

DT Online at 23:56, 25 February 2015

2015-02-25T23:56:01Z

← Older revision		Revision as of 23:56, 25 February 2015
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	* It can be derived from this that [http://www.mathopenref.com/arcradius.html '''finding the centre of an arch'''] doorway or window for example can be found by the formula: [[File:ArcRadius.png\|125px]]		* It can be derived from this that [http://www.mathopenref.com/arcradius.html '''finding the centre of an arch'''] doorway or window for example can be found by the formula: [[File:ArcRadius.png\|125px]]
	: Where: '''W''' is the length of the chord defining the base of the arc and '''H''' is the height measured at the midpoint of the arc's base.		: Where: '''W''' is the length of the chord defining the base of the arc and '''H''' is the height measured at the midpoint of the arc's base.
	* This is a particular case of the '''[[Intersecting Chords Theorem]]''' which states that ~~this~~ relationship ''(A x B = C x D)'' is true for <u> any</u> two ~~intesecting~~ chords, whether or not one of them is a diameter</span>		* This is a particular case of the '''[[Intersecting Chords Theorem]]''' which states that the relationship ''(A x B = C x D)'' is true for <u> any</u> two intersecting chords, whether or not one of them is a diameter</span>
	\| [[File:ChordsTheorem.png\|250px\|right]]		\| [[File:ChordsTheorem.png\|250px\|right]]
	\|-		\|-

DT Online at 23:51, 25 February 2015

2015-02-25T23:51:56Z

← Older revision		Revision as of 23:51, 25 February 2015
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	* Join the point to the circle centre and construct a semi-circle on this line.		* Join the point to the circle centre and construct a semi-circle on this line.
	* Where the semi-circle cuts the circumference provides the ''Point of Tangency'' to which may be drawn both the ''Normal'' and ''Tangent'' required.		* Where the semi-circle cuts the circumference provides the ''Point of Tangency'' to which may be drawn both the ''Normal'' and ''Tangent'' required.
	</span>		----
			<span style="color: green">'''Note:'''
			This concept is the basis of the design of '''[[Squares\|Centre Squares]]''' ''( or Centre Finders)''
			</span>
			----</span>
	\| [[File:TangentPointCircle.png\|250px\|right]]		\| [[File:TangentPointCircle.png\|250px\|right]]
	\|-
	\|}		\|}

DT Online at 23:48, 25 February 2015

2015-02-25T23:48:52Z

← Older revision		Revision as of 23:48, 25 February 2015
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	* The '''[[Basic Constructions\|perpendicular bisectors]]''' of <u>any</u> two chords to a circle will intersect at the centre of that circle		* The '''[[Basic Constructions\|perpendicular bisectors]]''' of <u>any</u> two chords to a circle will intersect at the centre of that circle
	* In practice, a 3rd chord is also bisected as shown to serve as an accuracy check when finding the circle centre.		* In practice, a 3rd chord is also bisected as shown to serve as an accuracy check when finding the circle centre.
	~~----~~
	~~<span style="color: green">'''Note:'''~~
	~~This concept is the basis of the design of '''[[Squares\|Centre Squares]]''' ''( or Centre Finders)''~~
	~~</span>~~
	~~----~~
	</span>		</span>
	\| [[File:IntersectingChords.png\|250px\|right]]		\| [[File:IntersectingChords.png\|250px\|right]]