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 formula_1. When two such triangles are placed in the positions shown in the illustration, the smallest rectangle that can enclose them has width formula_2 and height formula_1. 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) formula_4. The remaining space at the top of the rectangle is a right triangle with acute angles of 15° and 75° and sides of formula_5, formula_6, and formula_4. 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 formula_12, formula_13, and formula_4. Its legs are  each the hypotenuse of a 45°–45°–90° triangle, one with legs of length formula_15 and one with legs of length formula_1. The 15°–75°–90° triangle is the same as above.