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°. It is named after Douglas S. Ailles who was a high school teacher at Kipling Collegiate Institute in Toronto.

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 . 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 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. The 15°–75°–90° triangle is the same as above.