45-45-90三角形計算機｜直角二等辺三角形

A 45-45-90 triangle is the second of the two most important special right triangles in mathematics, alongside the 30-60-90 triangle. Named for its three interior angles of 45°, 45°, and 90°, this triangle is also called an isosceles right triangle because its two legs are always equal in length. The sides follow the exact ratio 1 : 1 : √2, where each leg has length 1 unit and the hypotenuse has length √2 ≈ 1.41421 units. As with the 30-60-90 triangle, this ratio scales perfectly: knowing any one side immediately determines all three sides. The 45-45-90 triangle arises from cutting a square along its diagonal. A square with side length 1 produces two congruent 45-45-90 triangles whose legs are the square's sides and whose hypotenuse is the square's diagonal. By the Pythagorean theorem, that diagonal equals √(1² + 1²) = √2. This simple construction confirms why the ratio is exact and why √2 — despite being irrational — is a fundamental constant in geometry. To find the hypotenuse from a known leg, multiply by √2 (approximately 1.41421356). To find a leg from the hypotenuse, divide by √2, equivalently multiply by √2/2 ≈ 0.70711. Because both legs are equal, knowing either leg gives the other immediately. The calculator performs these operations in full double-precision floating-point arithmetic, giving results accurate to at least 10 significant digits. In architecture and design, 45-45-90 triangles are ubiquitous. Roof trusses using a 45-degree pitch, corner bracing in wooden frames, and mitre cuts at 45 degrees all rely on this triangle. Tile layouts often use 45-degree diagonals, where understanding the relationship between tile size and diagonal length prevents wasteful cuts. In computer graphics and game development, 45-degree angles appear constantly in pixel art grids, isometric projections, and collision geometry. The trigonometric values for 45 degrees — sin(45°) = cos(45°) = √2/2 — come directly from this triangle and appear throughout calculus, signal processing, and physics. In physics, the 45-degree launch angle maximises projectile range on a flat surface; the horizontal and vertical components of velocity are equal at that angle, which is the direct consequence of a 45-45-90 triangle decomposition. In woodworking, a perfect mitre joint requires that two pieces be cut at exactly 45 degrees, and the diagonal of the resulting square corner is the hypotenuse of a 45-45-90 triangle. Everywhere that equal proportions and right angles combine — from picture frames to window sashes to the cuts made by a table saw at 45 degrees — the 45-45-90 triangle provides the governing geometry. This calculator automates the arithmetic so you can focus on the design or engineering task at hand.