Ailles rectangle

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] 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

From the construction of the rectangle, it follows that and An alternative construction (also by Ailles) places a 30°–60°–90° triangle in the middle with sidelengths of

Its legs are each the hypotenuse of a 45°–45°–90° triangle, one with legs of length

and one with legs of length