Thirty-sixty-ninety triangle solver

Those fixed angles force its short leg, long leg, and hypotenuse into the permanent ratio one to the square root of three to two. This solver accepts any one positive side, identifies its role, and calculates the complete set of side lengths with deterministic arithmetic. It is useful for checking geometry work, preparing dimensions, and building repeatable calculations without entering angles or choosing a general trigonometric formula. The browser version runs locally, while an API request costs $0.002.

Understand the special side ratio

Every thirty-sixty-ninety triangle is similar to every other triangle with the same three angles. Similarity is why one fixed ratio solves the entire family: short leg to long leg to hypotenuse equals one to the square root of three to two. The short leg is always opposite the thirty-degree angle, the long leg is opposite the sixty-degree angle, and the hypotenuse is opposite the ninety-degree angle. If the short leg is represented by x, the long leg is x multiplied by the square root of three and the hypotenuse is two x. This relationship can also be derived by cutting an equilateral triangle in half through its altitude. The cut creates two congruent right triangles, halves the original base, and leaves the altitude equal to root three times the half-base. Because the ratio depends on side roles, a bare number is not enough input. A length of ten could be the short leg, the long leg, or the hypotenuse, producing three differently scaled triangles. Always identify which side the measurement represents before applying a formula or using the solver.

Enter one side and interpret the result

Provide side as a positive finite number and set side_type to short, long, or hypotenuse. Choose short when the known side lies opposite the thirty-degree angle. Choose long when it lies opposite the sixty-degree angle. Choose hypotenuse for the side opposite the right angle, which is also the triangle's longest edge. The response returns all three named sides so the known value appears alongside the two calculated values. It also echoes the selected role in given and includes the ratio as a readable reminder of the method. When the short leg is known, the solver multiplies it by root three and by two. When the long leg is known, it first divides by root three to recover the short leg. When the hypotenuse is known, it first divides by two. Outputs are rounded to fifteen significant digits, which keeps ordinary decimal results stable while preserving useful precision across small and large accepted measurements. The solver does not attach physical units, so all returned sides use whatever consistent unit the input represents, whether centimeters, meters, inches, or an abstract coordinate scale.

Validate calculations and avoid common mistakes

A quick validation starts with ordering: for every positive input, the short leg must be smaller than the long leg, and the long leg must be smaller than the hypotenuse. Next, divide the long leg by the short leg; the result should be approximately 1.732, the square root of three. Dividing the hypotenuse by the short leg should give exactly two within displayed precision. You can also check the Pythagorean theorem: short squared plus long squared equals hypotenuse squared, because x squared plus three x squared equals four x squared. The most frequent mistake is assigning a side to the wrong role. A diagram may be rotated or reflected, so do not guess from whether an edge looks horizontal, vertical, left, or right; use the angle opposite it. Another mistake is multiplying when division is required. Given the long leg, divide by root three before doubling the recovered short leg. Given the hypotenuse, halve it first. Zero, negative, missing, nonnumeric, infinite, and out-of-range side values are rejected as invalid input rather than producing a degenerate or misleading triangle. Successful API requests use the base price $0.002.