Trig Triangle Calculator

The calculator relies on fundamental trigonometric relationships and the Pythagorean theorem to solve for missing variables. For any right-angled triangle with legs a and b, and hypotenuse c, the Pythagorean theorem states that c^2 = a^2 + b^2. Furthermore, trigonometric ratios define the relationships between angles and side lengths: the sine of an angle is the ratio of the opposite leg to the hypotenuse (sin(Α) = a/c), and the tangent is the ratio of the opposite leg to the adjacent leg (tan(Α) = a/b). When given alternative parameters such as total area and one leg, the calculator derives the missing altitude and hypotenuse by working backward through area formulas (Area = 0.5 × a × b).

Let us solve for the leg measurements of an isosceles right triangle (a 45-45-90 triangle) with a known hypotenuse of 18 cm. In a 45-45-90 triangle, both legs are equal in length (a = b). Using the Pythagorean theorem, c^2 = a^2 + a^2, which simplifies to c^2 = 2a^2. Substituting our hypotenuse: 18^2 = 2a^2, yielding 324 = 2a^2. Dividing by 2 gives a^2 = 162. Taking the square root, a = sqrt(162), which simplifies to 9√2 cm, or approximately 12.73 cm for each leg.

Always verify your input units before running the calculation to ensure all lengths use the same measurement standard (e.g., all centimeters or all inches). Remember that in a right-angled triangle, the hypotenuse is always the longest side; if your calculated hypotenuse is shorter than either leg, double-check your initial inputs. When dealing with angle inputs, confirm whether your calculator is set to degrees or radians to avoid drastic calculation errors.