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 15ð and 75ð. It is named after Douglas S. Ailles who was a high school teacher at Kipling Collegiate Institute in Toronto.
A 30ðâÂÂ60ðâÂÂ90ð 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 45ðâÂÂ45ðâÂÂ90ð 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 15ð and 75ð and sides of , , and .
From the construction of the rectangle, it follows that
and
An alternative construction (also by Ailles) places a 30ðâÂÂ60ðâÂÂ90ð triangle in the middle with sidelengths of , , and . Its legs are each the hypotenuse of a 45ðâÂÂ45ðâÂÂ90ð triangle, one with legs of length and one with legs of length . The 15ðâÂÂ75ðâÂÂ90ð triangle is the same as above.