{{Short description|Right triangle with a feature making calculations on the triangle easier}} {{Redirect2|45-45-90 triangle|30-60-90 triangle|the drawing tool|Set square}} [[Image:Euler diagram of triangle types.svg|thumb|320px|Position of some special triangles in an [[Euler diagram]] of types of triangles, using the definition that [[isosceles triangle]]s have at least two equal sides, i.e. that [[equilateral triangle]]s are isosceles]] A '''special right triangle''' is a [[right triangle]] with some notable feature that makes calculations on the [[triangle]] easier, or for which simple formulas exist.
The various relationships between the angles and sides of such triangles allow one to quickly calculate some useful quantities in [[geometry|geometric]] problems without resorting to more advanced methods.
==Angle-based== [[File:Special right triangles for trig.svg|right|thumb|Special angle-based triangles inscribed in a unit circle are handy for visualizing and remembering [[trigonometric functions]] of multiples of 30 and 45 degrees.]]
''Angle-based'' special right triangles are those involving some special relationship between the triangle's three angle measures. The angles of these triangles are such that the larger (right) angle, which is 90 [[degree (angle)|degrees]] or {{sfrac|{{pi}}|2}} [[radian]]s, is equal to the sum of the other two angles.
The side lengths of these triangles can be deduced based on the [[unit circle]], or with the use of other [[geometry|geometric]] methods; and these approaches may be extended to produce the values of [[trigonometric functions]] for some common angles, shown in the table below.
{| class="wikitable" ! [[Degree (angle)|degree]]s !! [[radian]]s !! [[Gon (angle)|gon]]s !! [[Turn (geometry)|turn]]s !! [[sine (trigonometric function)|sin]] !! [[cosine (trigonometric function)|cos]] !! [[tangent (trigonometric function)|tan]] !! [[cotangent (trigonometric function)|cotan]] |- | 0° || 0 || 0<sup>g</sup> || 0 || {{sfrac|{{sqrt|0}}|2}} = 0 || {{sfrac|{{sqrt|4}}|2}} = 1 || 0 || undefined |- | 30° || {{sfrac|{{pi}}|6}} || {{sfrac|33|1|3}}<sup>g</sup> || {{sfrac|1|12}} || {{sfrac|{{sqrt|1}}|2}} = {{sfrac|1|2}} || {{sfrac|{{sqrt|3}}|2}} || {{sfrac|1|{{radic|3}}}} || {{sqrt|3}} |- | 45° || {{sfrac|{{pi}}|4}} || 50<sup>g</sup> || {{sfrac|1|8}} || {{sfrac|{{sqrt|2}}|2}} = {{sfrac|1|{{radic|2}}}} || {{sfrac|{{sqrt|2}}|2}} = {{sfrac|1|{{radic|2}}}} || 1 || 1 |- | 60° || {{sfrac|{{pi}}|3}} || {{sfrac|66|2|3}}<sup>g</sup> || {{sfrac|1|6}} || {{sfrac|{{sqrt|3}}|2}} || {{sfrac|{{sqrt|1}}|2}} = {{sfrac|1|2}} || {{sqrt|3}} || {{sfrac|1|{{radic|3}}}} |- | 90° || {{sfrac|{{pi}}|2}} || 100<sup>g</sup> || {{sfrac|1|4}} || {{sfrac|{{sqrt|4}}|2}} = 1 || {{sfrac|{{sqrt|0}}|2}} = 0 || undefined || 0 |}
{{multiple image | width = 120 | image1 = Tile V488 bicolor.svg | caption1 = 45°–45°–90° | image2 = Tile V46b.svg | caption2 = 30°–60°–90° }}
The 45°–45°��90° triangle, the 30°–60°–90° triangle, and the [[equilateral triangle|equilateral]]/equiangular (60°–60°–60°) triangle are the three [[Möbius triangle]]s in the plane, meaning that they [[tessellate]] the plane via [[reflection (mathematics)|reflections]] in their sides; see [[Triangle group]]. ==={{anchor|45-45-90 triangle|90-45-45 triangle}}45°–45°–90° triangle=== [[File:Squadra 45.jpg|thumb|x150px|left|[[Set square]] shaped as 45°–45°–90° triangle]] [[Image:45-45-triangle.svg|thumb|right|x150px|The side lengths of a 45°–45°–90° triangle]] [[File:45° 45° 90° Special Right Triangle.svg|thumb|45°–45°–90° [[right triangle]] of [[hypotenuse]] length 1]]
In [[plane geometry]], dividing a [[square]] along its diagonal results in two '''isosceles right triangles''', each with one right angle (90°, {{sfrac|{{pi}}|2}} radians) and two other congruent angles each measuring half of a right angle (45°, or {{sfrac|{{pi}}|4}} radians). The sides in this triangle are in the ratio 1 : 1 : {{sqrt|2}}, which follows immediately from the [[Pythagorean theorem]].
Of all right triangles, such 45°–45°–90° degree triangles have the smallest ratio of the [[hypotenuse]] to the sum of the legs, namely {{sfrac|{{sqrt|2}}|2}}.<ref name=PL>Posamentier, Alfred S., and Lehman, Ingmar. ''[[The Secrets of Triangles]]''. Prometheus Books, 2012.</ref>{{rp|p. 282, p. 358}} and the greatest ratio of the [[altitude (triangle)|altitude]] from the hypotenuse to the sum of the legs, namely {{sfrac|{{sqrt|2}}|4}}.<ref name=PL/>{{rp|p.282}}
Triangles with these angles are the only possible right triangles that are also [[isosceles triangle]]s in [[Euclidean geometry]]. However, in [[spherical geometry]] and [[hyperbolic geometry]], there are infinitely many different shapes of right isosceles triangles.
==={{anchor|30-60-90 triangle|90-60-30 triangle}}30°–60°–90° triangle=== [[File:Squadra 30 60.jpg|thumb|x150px|left|Set square, shaped as 30°–60°–90° triangle]] [[File:Equilateral triangle with height square root of 3.svg|alt=|thumb|x141px|The side lengths of a 30°–60°–90° triangle]] [[File:30° 60° 90° Special Right Triangle.svg|thumb|30°–60°–90° [[right triangle]] of [[hypotenuse]] length 1]] Another type of special right triangle is the 30°-60°-90° triangle, which refers to any triangle with those three angle measures. Notably, these angles are in the ratio 1 : 2 : 3.
A useful property of such triangles is that their side lengths are in the ratio 1 : [[Square root of 3|{{sqrt|3}}]] : 2. This property can be proven using [[trigonometry]], or via the [[geometry|geometric]] proof below:
Draw an equilateral triangle ''ABC'' with side length 2, and with point ''M'' as the midpoint of segment ''BC''. Draw an altitude line from ''A'' to ''M''. Then ''ABM'' is a 30°–60°–90° triangle with hypotenuse of length 2, and base ''BM'' of length 1.
The fact that the remaining leg ''AM'' has length [[Square root of 3|{{sqrt|3}}]] follows immediately from the [[Pythagorean theorem]].
The 30°–60°–90° triangle is the only right triangle whose angles are in an [[arithmetic progression]]. The proof of this fact is simple and follows on from the fact that if ''α'', {{nowrap|''α'' + ''δ''}}, {{nowrap|''α'' + 2''δ''}} are the angles in the progression then the sum of the angles {{nowrap|3''α'' + 3''δ''}} = 180°. After dividing by 3, the angle {{nowrap|''α'' + ''δ''}} must be 60°. The right angle is 90°, leaving the remaining angle to be 30°.
==Side-based== Right triangles whose sides are of [[integer]] lengths, with the sides collectively known as [[Pythagorean triple]]s, possess angles that cannot all be [[rational numbers]] of [[Degree (angle)|degrees]].<ref>{{Cite journal|title=Rational Triangle|last=Weisstein|first=Eric W|journal=MathWorld|url=http://mathworld.wolfram.com/RationalTriangle.html}}</ref> (This follows from [[Niven's theorem]].) They are most useful in that they may be easily remembered and any [[multiple (mathematics)|multiple]] of the sides produces the same relationship. Using Euclid's formula for generating Pythagorean triples, the sides must be in the ratio
:{{nowrap|''m''{{sup|2}} − ''n''{{sup|2}} : 2''mn'' : ''m''{{sup|2}} + ''n''{{sup|2}}}}
where ''m'' and ''n'' are any positive integers such that {{nowrap|''m'' > ''n''}}.
===Common Pythagorean triples=== {{Main|Pythagorean triple}}
There are several Pythagorean triples which are well-known, including those with sides in the ratios:
:{| border="0" cellpadding="1" cellspacing="0" |align="right"|3 :||align="right"| 4 :||align="right"| 5 |- |align="right"|5 :||align="right"|12 :||align="right"|13 |- |align="right"|8 :||align="right"|15 :||align="right"|17 |- |align="right"|7 :||align="right"|24 :||align="right"|25 |- |align="right"|9 :||align="right"|40 :||align="right"|41 |}
The 3 : 4 : 5 triangles are the only right triangles with edges in [[arithmetic progression]]. Triangles based on Pythagorean triples are [[Heronian triangle|Heronian]], meaning they have integer [[area]] as well as integer sides.
The possible use of the 3 : 4 : 5 triangle in [[Ancient Egypt]], with the supposed use of a knotted rope to lay out such a triangle, and the question whether Pythagoras' theorem was known at that time, have been much debated.<ref name=Cooke2011>{{cite book |last=Cooke |first=Roger L. |title=The History of Mathematics: A Brief Course |url=https://books.google.com/books?id=wOGh7XPowAMC |edition=2nd |year=2011|publisher=John Wiley & Sons |isbn=978-1-118-03024-0 |pages=237–238}}</ref> It was first conjectured by the historian [[Moritz Cantor]] in 1882.<ref name=Cooke2011/> It is known that right angles were laid out accurately in Ancient Egypt; that their surveyors did use ropes for measurement;<ref name=Cooke2011/> that [[Plutarch]] recorded in ''[[De Iside et Osiride|Isis and Osiris]]'' (around 100 AD) that the Egyptians admired the 3 : 4 : 5 triangle;<ref name=Cooke2011/> and that the [[Berlin Papyrus 6619]] from the [[Middle Kingdom of Egypt]] (before 1700 BC) stated that "the area of a square of 100 is equal to that of two smaller squares. The side of one is {{sfrac|1|2}} + {{sfrac|1|4}} the side of the other."<ref>{{cite book |author=Gillings, Richard J. |title=Mathematics in the Time of the Pharaohs |url=https://archive.org/details/mathematicsintim0000gill |url-access=registration |publisher=Dover |date=1982 |page=[https://archive.org/details/mathematicsintim0000gill/page/161 161]}}</ref> The historian of mathematics Roger L. Cooke observes that "It is hard to imagine anyone being interested in such conditions without knowing the Pythagorean theorem."<ref name=Cooke2011/> Against this, Cooke notes that no Egyptian text before 300 BC actually mentions the use of the theorem to find the length of a triangle's sides, and that there are simpler ways to construct a right angle. Cooke concludes that Cantor's conjecture remains uncertain: he guesses that the Ancient Egyptians probably did know the Pythagorean theorem, but that "there is no evidence that they used it to construct right angles".<ref name=Cooke2011/>
The following are all the Pythagorean triple ratios expressed in lowest form (beyond the five smallest ones in lowest form in the list above) with both non-hypotenuse sides less than 256:
:{| border="0" cellpadding="1" cellspacing="0" align="left" style="margin-right: 2em" |align="right"|11 :||align="right"| 60 :||align="right"| 61 |- |align="right"|12 :||align="right"| 35 :||align="right"| 37 |- |align="right"|13 :||align="right"| 84 :||align="right"| 85 |- |align="right"|15 :||align="right"|112 :||align="right"|113 |- |align="right"|16 :||align="right"| 63 :||align="right"| 65 |- |align="right"|17 :||align="right"|144 :||align="right"|145 |- |align="right"|19 :||align="right"|180 :||align="right"|181 |- |align="right"|20 :||align="right"| 21 :||align="right"| 29 |- |align="right"|20 :||align="right"| 99 :||align="right"|101 |- |align="right"|21 :||align="right"|220 :||align="right"|:221 |}
{| border="0" cellpadding="1" cellspacing="0" align="left" style="margin-right: 2em" |align="right"| 24 :||align="right"|143 :||align="right"|145 |- |align="right"| 28 :||align="right"| 45 :||align="right"| 53 |- |align="right"| 28 :||align="right"|195 :||align="right"|197 |- |align="right"| 32 :||align="right"|255 :||align="right"|257 |- |align="right"| 33 :||align="right"| 56 :||align="right"| 65 |- |align="right"| 36 :||align="right"| 77 :||align="right"| 85 |- |align="right"| 39 :||align="right"| 80 :||align="right"| 89 |- |align="right"| 44 :||align="right"|117 :||align="right"|125 |- |align="right"| 48 :||align="right"| 55 :||align="right"| 73 |- |align="right"| 51 :||align="right"|140 :||align="right"|149 |}
{| border="0" cellpadding="1" cellspacing="0" align="left" style="margin-right: 2em" |align="right"| 52 :||align="right"|165 :||align="right"|173 |- |align="right"| 57 :||align="right"|176 :||align="right"|185 |- |align="right"| 60 :||align="right"| 91 :||align="right"|109 |- |align="right"| 60 :||align="right"|221 :||align="right"|229 |- |align="right"| 65 :||align="right"| 72 :||align="right"| 97 |- |align="right"| 84 :||align="right"|187 :||align="right"|205 |- |align="right"| 85 :||align="right"|132 :||align="right"|157 |- |align="right"| 88 :||align="right"|105 :||align="right"|137 |- |align="right"| 95 :||align="right"|168 :||align="right"|193 |- |align="right"| 96 :||align="right"|247 :||align="right"|265 |}
{| border="0" cellpadding="1" cellspacing="0" |align="right"|104 :||align="right"|153 :||align="right"|185 |- |align="right"|105 :||align="right"|208 :||align="right"|233 |- |align="right"|115 :||align="right"|252 :||align="right"|277 |- |align="right"|119 :||align="right"|120 :||align="right"|169 |- |align="right"|120 :||align="right"|209 :||align="right"|241 |- |align="right"|133 :||align="right"|156 :||align="right"|205 |- |align="right"|140 :||align="right"|171 :||align="right"|221 |- |align="right"|160 :||align="right"|231 :||align="right"|281 |- |align="right"|161 :||align="right"|240 :||align="right"|289 |- |align="right"|204 :||align="right"|253 :||align="right"|325 |- |align="right"|207 :||align="right"|224 :||align="right"|305 |} {{clear}}
===Almost-isosceles Pythagorean triples=== Isosceles right-angled triangles cannot have sides with integer values, because the ratio of the hypotenuse to either other side is {{sqrt|2}} and [[Square root of 2#Proofs of irrationality|{{sqrt|2}} cannot be expressed as a ratio of two integers]]. However, infinitely many ''almost-isosceles'' right triangles do exist. These are right-angled triangles with integer sides for which the lengths of the [[Cathetus|non-hypotenuse edges]] differ by one.<ref>{{citation | last1 = Forget | first1 = T. W. | last2 = Larkin | first2 = T. A. | issue = 3 | journal = Fibonacci Quarterly | pages = 94–104 | title = Pythagorean triads of the form ''x'', ''x'' + 1, ''z'' described by recurrence sequences | url = http://www.fq.math.ca/Scanned/6-3/6-3/forget.pdf | volume = 6 | year = 1968| doi = 10.1080/00150517.1968.12431232 }}.</ref><ref>{{citation | last1 = Chen | first1 = C. C. | last2 = Peng | first2 = T. A. | journal = The Australasian Journal of Combinatorics | mr = 1327342 | pages = 263–267 | title = Almost-isosceles right-angled triangles | url = http://ajc.maths.uq.edu.au/pdf/11/ajc-v11-p263.pdf | volume = 11 | year = 1995}}.</ref> Such almost-isosceles right-angled triangles can be obtained recursively,
:''a''<sub>0</sub> = 1, ''b''<sub>0</sub> = 2 :''a''<sub>''n''</sub> = 2''b''<sub>''n''−1</sub> + ''a''<sub>''n''−1</sub> :''b''<sub>''n''</sub> = 2''a''<sub>''n''</sub> + ''b''<sub>''n''−1</sub>
''a''<sub>''n''</sub> is length of hypotenuse, ''n'' = 1, 2, 3, .... Equivalently,
:<math>(\tfrac{x-1}{2})^2+(\tfrac{x+1}{2})^2 = y^2</math>
where {''x'', ''y''} are solutions to the [[Pell equation]] {{nowrap|''x''{{sup|2}} − 2''y''{{sup|2}} {{=}} −1}}, with the hypotenuse ''y'' being the odd terms of the [[Pell numbers]] '''1''', 2, '''5''', 12, '''29''', 70, '''169''', 408, '''985''', 2378... {{OEIS|id=A000129}}.. The smallest Pythagorean triples resulting are:<ref>{{OEIS|A001652}}</ref>
:{| border="0" cellpadding="1" cellspacing="0" align="left" style="margin-right: 3em" |align="right"| 3 :||align="right"| 4 :||align="right"| 5 |- |align="right"| 20 :||align="right"| 21 :||align="right"| 29 |- |align="right"| 119 :||align="right"| 120 :||align="right"| 169 |- |align="right"| 696 :||align="right"| 697 :||align="right"| 985 |}
{| border="0" cellpadding="1" cellspacing="0" align="left" |align="right"| 4,059 :||align="right"| 4,060 :||align="right"| 5,741 |- |align="right"| 23,660 :||align="right"| 23,661 :||align="right"| 33,461 |- |align="right"| 137,903 :||align="right"| 137,904 :||align="right"| 195,025 |- |align="right"| 803,760 :||align="right"| 803,761 :||align="right"| 1,136,689 |} {{clear}}
Alternatively, the same triangles can be derived from the [[square triangular number]]s.<ref>{{citation | last = Nyblom | first = M. A. | issue = 4 | journal = The Fibonacci Quarterly | mr = 1640364 | pages = 319–322 | title = A note on the set of almost-isosceles right-angled triangles | url = http://www.fq.math.ca/Scanned/36-4/nyblom.pdf | volume = 36 | year = 1998| doi = 10.1080/00150517.1998.12428915 }}.</ref>
===Arithmetic and geometric progressions=== [[File:Kepler triangle.svg|right|thumb|A '''Kepler triangle''' is a right triangle formed by three squares with areas in geometric progression according to the '''[[golden ratio]]'''.]] {{Main article|Kepler triangle}}
The Kepler triangle is a right triangle whose sides are in [[geometric progression]]. If the sides are formed from the geometric progression ''a'', ''ar'', ''ar''<sup>2</sup> then its common ratio ''r'' is given by ''r'' = {{sqrt|''φ''}} where ''φ'' is the [[golden ratio]]. Its sides are therefore in the ratio {{nowrap|1 : {{sqrt|''φ''}} : ''φ''}}. Thus, the shape of the Kepler triangle is uniquely determined (up to a scale factor) by the requirement that its sides be in geometric progression.
The 3–4–5 triangle is the unique right triangle (up to scaling) whose sides are in [[arithmetic progression]].<ref>{{citation | last1 = Beauregard | first1 = Raymond A. | last2 = Suryanarayan | first2 = E. R. | doi = 10.2307/2691431 | issue = 2 | journal = [[Mathematics Magazine]] | mr = 1448883 | pages = 105–115 | title = Arithmetic triangles | volume = 70 | year = 1997| jstor = 2691431 }}.</ref>
===Sides of regular polygons=== [[File:Euclid XIII.10.svg|thumb|The sides of a pentagon, hexagon, and decagon, inscribed in [[Congruence (geometry)|congruent]] circles, form a right triangle.]] Let <math display=block>a=2\sin\frac{\pi}{10}=\frac{-1+\sqrt5}{2}=\frac1\varphi\approx 0.618</math> be the side length of a regular [[decagon]] inscribed in the unit circle, where <math>\varphi</math> is the golden ratio. Let <math display=block>b=2\sin\frac{\pi}{6}=1</math> be the side length of a regular [[hexagon]] in the unit circle, and let <math display=block>c=2\sin\frac{\pi}{5}=\sqrt{\frac{5-\sqrt5}{2}}\approx 1.176</math> be the side length of a regular [[pentagon]] in the unit circle. Then <math>a^2+b^2=c^2</math>, so these three lengths form the sides of a right triangle.<ref>[http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII10.html Euclid's ''Elements'', Book XIII, Proposition 10].</ref> The same triangle forms half of a [[golden rectangle]]. It may also be found within a [[regular icosahedron]] of side length <math>c</math>: the shortest line segment from any vertex <math>V</math> to the plane of its five neighbors has length <math>a</math>, and the endpoints of this line segment together with any of the neighbors of <math>V</math> form the vertices of a right triangle with sides <math>a</math>, <math>b</math>, and <math>c</math>.<ref>[http://ncatlab.org/nlab/show/pentagon+decagon+hexagon+identity nLab: pentagon decagon hexagon identity].</ref>
==See also== * [[Ailles rectangle]], combining several special right triangles * [[Integer triangle]] * [[Spiral of Theodorus]]
==References== <references/>
==External links== * [https://www.mathopenref.com/triangle345.html 3 : 4 : 5 triangle] * [https://www.mathopenref.com/triangle306090.html 30–60–90 triangle] * [https://www.mathopenref.com/triangle454590.html 45–45–90 triangle]{{snd}} with interactive animations
{{DEFAULTSORT:Special Right Triangles}} [[Category:Euclidean plane geometry]] [[Category:Types of triangles]]