Special Right Triangles Calculator

Special right triangles have fixed angle measures and therefore fixed side ratios. Once one side is known, every other side follows without using inverse trigonometric functions. The two patterns encountered most often are the 30-60-90 triangle and the 45-45-90 triangle. Both include a 90-degree angle, so the side opposite that angle is always the hypotenuse and is always the longest side. In a 30-60-90 triangle, the sides opposite 30, 60, and 90 degrees occur in the ratio 1 to square root of 3 to 2. The shortest leg lies opposite the 30-degree angle. If that leg has length s, the longer leg is s times square root of 3 and the hypotenuse is 2s. This calculator treats the entered leg as that short leg, then applies the exact ratio. The relationship also appears when an equilateral triangle is divided in half along an altitude. In a 45-45-90 triangle, both acute angles are equal, so the two legs have equal length. If either leg has length s, the hypotenuse is s times square root of 2. This pattern is also called an isosceles right triangle. It appears in square diagonals: a square with side s has diagonal s times square root of 2 because the diagonal divides the square into two congruent 45-45-90 triangles. The displayed decimals are rounded to six places for readability, but the internal calculation uses the full values of square root of 2 and square root of 3. In exact work, it is often preferable to leave answers in radical form, such as 5 square root of 3. Decimal values are convenient for measurement, construction, and checking homework, while radical values preserve exactness for later algebraic operations. These ratios can also be verified with the Pythagorean theorem. For a 30-60-90 triangle, the squares of s and s times square root of 3 add to 4s squared, which is the square of 2s. For a 45-45-90 triangle, two copies of s squared add to 2s squared, matching the square of s times square root of 2. Use the calculator to obtain quick dimensions, then keep the angle pattern and unit consistency in mind when applying the result.