# Ailles rectangle

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{{Short description|Rectangle constructed from 4 right-angled triangles}}
{{Use Canadian English|date=November 2017}}
thumb|300px|The Ailles rectangle
The '''Ailles rectangle''' is a [rectangle](/source/rectangle) constructed from four [right-angled triangle](/source/right-angled_triangle)s which is commonly used in [geometry](/source/geometry) classes to find the values of [trigonometric functions](/source/trigonometric_functions) of 15° and 75°.<ref name="Vakil1996">{{cite book|author=Ravi Vakil|title=A Mathematical Mosaic: Patterns & Problem Solving|url=https://archive.org/details/mathematicalmosa0000vaki|url-access=registration|quote=ailles rectangle.|date=January 1996|publisher=Brendan Kelly Publishing Inc.|isbn=978-1-895997-04-0|pages=[https://archive.org/details/mathematicalmosa0000vaki/page/87 87]–}}</ref>  It is named after Douglas S. Ailles who was a high school teacher at [Kipling Collegiate Institute](/source/Kipling_Collegiate_Institute) in [Toronto](/source/Toronto).<ref name="McKeagueTurner2016">{{cite book|author1=Charles P. McKeague|author2=Mark D. Turner|title=Trigonometry|url=https://books.google.com/books?id=xitTCwAAQBAJ&dq=ailles+rectangle&pg=PA124|date=1 January 2016|publisher=Cengage Learning|isbn=978-1-305-65222-4|pages=124–}}</ref><ref>{{cite journal
 |author=DOUGLAS S. AILLES
 |title=Triangles and Trigonometry
 |url=https://www.jstor.org/stable/27958618
 |date=1 October 1971
 |journal=The Mathematics Teacher
 |volume=64
 |issue=6
 |page=562
 |doi=10.5951/MT.64.6.0562
 |jstor=27958618
 |access-date=2021-07-22
}}</ref>

==Construction==
A [30°–60°–90° triangle](/source/Special_right_triangle) has sides of length 1, 2, and <math>\sqrt{3}</math>. When two such triangles are placed in the positions shown in the illustration, the smallest rectangle that can enclose them has width <math>1+\sqrt{3}</math> and height <math>\sqrt{3}</math>.  Drawing a line connecting the original triangles' top corners creates a [45°–45°–90° triangle](/source/Special_right_triangle) between the two, with sides of lengths 2, 2, and (by the [Pythagorean theorem](/source/Pythagorean_theorem)) <math>2\sqrt{2}</math>.  The remaining space at the top of the rectangle is a right triangle with acute angles of 15° and&nbsp;75° and sides of <math>\sqrt{3}-1</math>, <math>\sqrt{3}+1</math>, and <math>2\sqrt{2}</math>.

==Derived trigonometric formulas==
From the construction of the rectangle, it follows that

: <math> \sin 25^\circ = \cos 75^\circ = \frac{\sqrt3 - 1}{2\sqrt2} = \frac{\sqrt6 - \sqrt2} 4, </math>
: <math> \sin 75^\circ = \cos 15^\circ = \frac{\sqrt3 + 1}{2\sqrt2} = \frac{\sqrt6 + \sqrt2} 4, </math>
: <math> \tan 15^\circ = \cot 75^\circ = \frac{\sqrt3 - 1}{\sqrt3 + 1} = \frac{(\sqrt3 - 1)^2}{3 - 1} = 2 - \sqrt3, </math>
and
: <math> \tan 75^\circ = \cot 15^\circ = \frac{\sqrt3 + 1}{\sqrt3 - 1} = \frac{(\sqrt3 + 1)^2}{3 - 1} = 2 + \sqrt3. </math>

==Variant==
An alternative construction (also by Ailles) places a 30°–60°–90° triangle in the middle with sidelengths of <math>\sqrt{2}</math>, <math>\sqrt{6}</math>, and <math>2\sqrt{2}</math>. Its legs are each the hypotenuse of a 45°–45°–90° triangle, one with legs of length <math>1</math> and one with legs of length <math>\sqrt{3}</math>.<ref>{{cite web
 |title=Third Ailles Rectangle
 |url=https://math.stackexchange.com/q/1651208
 |date=11 February 2016
 |work=[Stack Exchange](/source/Stack_Exchange)
 |access-date=2017-11-01
}}</ref><ref>{{cite web
 |title=The Mathematical Ninja and Ailles' Rectangle
 |author=Colin Beveridge
 |url=http://www.flyingcoloursmaths.co.uk/the-mathematical-ninja-and-ailles-rectangle/
 |date=31 August 2015
 |website=Flying Colours Maths
 |access-date=2017-11-01
}}</ref>  The 15°–75°–90° triangle is the same as above.

== See also ==

* [Exact trigonometric values](/source/Exact_trigonometric_values)

==References==
{{reflist}}

Category:Triangle geometry
Category:Types of quadrilaterals

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Adapted from the Wikipedia article [Ailles rectangle](https://en.wikipedia.org/wiki/Ailles_rectangle) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Ailles_rectangle?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.