45-45-90 Triangle Calculator

The 45-45-90 triangle is an isosceles right triangle: two legs are equal and the angles are 45, 45, and 90 degrees. The side ratios 1 : 1 : sqrt(2) follow directly from the Pythagorean theorem: if each leg has length s, then the hypotenuse is sqrt(s^2 + s^2) = s*sqrt(2). The exact trigonometric values follow: sin(45) = cos(45) = 1/sqrt(2) = sqrt(2)/2, and tan(45) = 1. The fact that tan(45) = 1 means a 45-degree line has slope 1, which is the diagonal direction. The 45-45-90 triangle arises naturally from the diagonal of a square. If a square has side length s, its diagonal has length s*sqrt(2), and the diagonal divides the square into two congruent 45-45-90 triangles. This is why TV screens are measured diagonally: a 50-inch diagonal screen with 16:9 aspect ratio has specific width and height determined by the relationship between diagonal and sides. In coordinate geometry, the 45-45-90 triangle appears in rotation by 45 degrees. The rotation matrix for 45 degrees uses cos(45) and sin(45), both equal to sqrt(2)/2. In physics, when forces or velocities act at 45 degrees, their horizontal and vertical components are equal, each being the magnitude times sqrt(2)/2. The irrationality of sqrt(2) was one of the first mathematical discoveries of the ancient Greeks and is attributed to the Pythagorean school. The proof that sqrt(2) is irrational (by contradiction, assuming it equals p/q in lowest terms) was a foundational moment in the history of mathematics, revealing that not all lengths are expressible as ratios of whole numbers.