The Ailles rectangle is a rectangle constructed from four right-angled triangles which is commonly used in geometry classes to find the values of trigonometric functions of 15° and 75°.[1] It is named after Douglas S. Ailles who was a high school teacher at Kipling Collegiate Institute in Toronto.[2][3]
![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Ailles_rectangle.png/300px-Ailles_rectangle.png)
Construction
A 30°–60°–90° triangle has sides of length 1, 2, and . When two such triangles are placed in the positions shown in the illustration, the smallest rectangle that can enclose them has width
and height
. Drawing a line connecting the original triangles' top corners creates a 45°–45°–90° triangle between the two, with sides of lengths 2, 2, and (by the Pythagorean theorem)
. The remaining space at the top of the rectangle is a right triangle with acute angles of 15° and 75° and sides of
,
, and
.
Derived trigonometric formulas
From the construction of the rectangle, it follows that
and
Variant
An alternative construction (also by Ailles) places a 30°–60°–90° triangle in the middle with sidelengths of ,
, and
. Its legs are each the hypotenuse of a 45°–45°–90° triangle, one with legs of length
and one with legs of length
.[4][5] The 15°–75°–90° triangle is the same as above.