45-45-90 Triangle Calculator

It is the isosceles right triangle: one 90-degree angle, two 45-degree acute angles, and two legs of equal length. Because two sides match, it is isosceles; because one angle is 90 degrees, it is also a right triangle. That single label, isosceles right triangle, fully describes the shape up to scale. The angles are fixed at 45, 45, and 90, so every 45-45-90 is a scaled copy of every other one, which is exactly what makes it a special right triangle worth memorizing.

Apply the Pythagorean theorem with a = b: c^2 = a^2 + a^2 = 2 * a^2, so c = sqrt(2 * a^2) = a * sqrt(2). That is where the 1 : 1 : sqrt(2) ratio comes from, and why sqrt(2) (about 1.41421356) shows up every time you scale the triangle. Pick any leg length, multiply by sqrt(2), and you have the hypotenuse exactly. The same identity, inverted, lets you recover the leg from the hypotenuse: leg = hypotenuse / sqrt(2), which is what the from-hypotenuse mode does.

Cut a square along its diagonal. A square has four 90-degree angles and four equal sides; one diagonal splits it into two congruent 45-45-90 triangles where the diagonal is the shared hypotenuse and the two square sides become the legs. That is the cleanest way to see why the legs are equal and why the diagonal of a unit square is sqrt(2). Carpenters, drafters, and quilters use this trick constantly because anywhere you have a square sheet of material, you have two perfect 45-45-90 triangles for free.

Both are special right triangles with fixed angle sets, but the ratios are different. A 45-45-90 has angles 45, 45, 90 and side ratio 1 : 1 : sqrt(2), so the two legs are equal. A 30-60-90 has angles 30, 60, 90 and side ratio 1 : sqrt(3) : 2, so the shorter leg (opposite 30) is half the hypotenuse, the longer leg (opposite 60) is the shorter leg times sqrt(3), and no two sides are equal. Rule of thumb: 45-45-90 = the half-square, 30-60-90 = the half-equilateral-triangle.

It appears wherever a 45-degree line meets a square corner. Mitre joints split a 90-degree corner into two 45-degree cuts, and a square's diagonal forms two 45-45-90 triangles. Drafting uses a 45-degree set square for perpendicular and 45-degree lines. Isometric axes are normally drawn with a 30-60-90 set square instead.