30 60 90 Triangle Calculator

A 30 60 90 triangle is a special right triangle with interior angles measuring 30 degrees, 60 degrees, and 90 degrees. Because of these specific angle measures, the side lengths always maintain a strict, predictable ratio of 1 to the square root of 3 to 2. If we let the shortest leg (opposite the 30-degree angle) be represented by 'a', then the longer leg (opposite the 60-degree angle) is equal to a * sqrt(3), and the hypotenuse (opposite the 90-degree angle) is equal to 2 * a. From these base dimensions, the total area can be computed using the formula Area = (a^2 * sqrt(3)) / 2, and the perimeter is found by summing all three sides: Perimeter = a + a * sqrt(3) + 2 * a.

Let us walk through a practical calculation where the shortest leg (A side) is given as 5 units. First, we find the B side (longer leg) by multiplying the A side by the square root of 3: b_side = 5 * 1.732 = 8.66 units. Next, we determine the C side (hypotenuse) by multiplying the A side by 2: c_side = 2 * 5 = 10 units. To find the area, we square the A side, multiply by the square root of 3, and divide by 2: area = (25 * 1.732) / 2 = 21.65 square units. Finally, the perimeter is calculated by adding all sides together: perimeter = 5 + 8.66 + 10 = 23.66 units.

Always identify which side corresponds to the 30-degree angle before beginning your manual calculations, as this shortest leg serves as the foundational multiplier for all other measurements. When working with decimal approximations of the square root of 3 (approximately 1.732), maintain adequate precision through your intermediate steps to prevent compounding rounding errors in your final answer. If you are given the hypotenuse instead of a leg, remember that you must divide by 2 to find the shortest leg first before applying any other ratios.