Special Right Triangle Calculator
Solve 45-45-90 and 30-60-90 triangles from one known side. Get all sides and the area.
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All Sides and Area
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Special Right Triangle Rules
45-45-90: legs are equal, hypotenuse = leg x sqrt(2)
30-60-90: long leg = short leg x sqrt(3), hypotenuse = 2 x short leg
Worked examples: a 45-45-90 triangle with leg 5 has hypotenuse 5 x 1.414 = 7.071 and area 5 x 5 / 2 = 12.5. A 30-60-90 triangle with short leg 4 has long leg 4 x 1.732 = 6.928, hypotenuse 2 x 4 = 8, and area 4 x 6.928 / 2 = 13.856.
Frequently Asked Questions
A special right triangle calculator solves 45-45-90 and 30-60-90 triangles. Enter one known side and it works out the other two sides and the area using the fixed side ratios of these triangles.
In a 45-45-90 triangle the two legs are equal and the hypotenuse is the leg times the square root of 2. With a leg of 5, the hypotenuse is 5 x 1.414 = 7.071 and the area is 12.5.
In a 30-60-90 triangle the short leg is opposite the 30 degree angle, the long leg is the short leg times the square root of 3, and the hypotenuse is twice the short leg. With a short leg of 4, the long leg is 6.928, the hypotenuse is 8, and the area is 13.856.
For a 45-45-90 triangle with leg a, the area is a squared divided by 2. For a 30-60-90 triangle with short leg a, the area is a squared times the square root of 3, divided by 2.
Enter any side you know and pick which side it is from the list. The tool works out the other two sides from the fixed ratios.
Their angles are fixed, so their side ratios are always the same. That means one known side is enough to find every other side, with no angles to measure.
Yes. The tool shows every side and the area, and the formula card shows the ratios used, so you can check each step.
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