Enter any 2 known values from the sides and angles to solve the right triangle. The calculator finds all remaining sides, both acute angles, area, and perimeter, then shows the steps used.
Leave the values you want to find blank. Angles are in degrees. At least one side is required alongside any angle.
A right triangle has one 90-degree angle (angle C) and two acute angles A and B. The side opposite the right angle is the hypotenuse c; the other two sides are legs a and b.
Given two legs, the hypotenuse is c = sqrt(a squared + b squared). Given one leg and the hypotenuse, the missing leg is a = sqrt(c squared minus b squared). The basic trig ratios tie each side pair to an angle:
Angle B always equals 90 degrees minus A because the angles of any triangle sum to 180 degrees and C is fixed at 90.
This is the classic 3-4-5 Pythagorean triple. For the underlying theorem, see Pythagorean Theorem Calculator. To find the slope of a line defined by two points on a right triangle, see Slope Calculator.
For an academic reference on right triangle trigonometry, see Wolfram MathWorld on right triangles.
Use the Pythagorean theorem: a squared + b squared = c squared. If you know both legs, c = sqrt(a squared + b squared). If you know one leg and the hypotenuse, the other leg is sqrt(c squared minus the known leg squared). This calculator handles all three cases automatically.
Apply an inverse trig function. With the opposite side and hypotenuse, angle A = arcsin(opposite / hypotenuse). With opposite and adjacent legs, angle A = arctan(opposite / adjacent). The remaining acute angle is 90 degrees minus A.
Area = (1/2) times leg a times leg b. The two legs serve as base and height because they meet at the right angle, so no separate altitude is needed.
The 3-4-5 and 5-12-13 Pythagorean triples come up constantly in geometry. Any whole-number multiple (6-8-10, 10-24-26) also forms a right triangle. The 45-45-90 triangle has legs in ratio 1:1 with hypotenuse = leg times sqrt(2); the 30-60-90 has sides in ratio 1 : sqrt(3) : 2.
For angle A opposite leg a: sin A = a / c; cos A = b / c; tan A = a / b. SOH-CAH-TOA is the standard memory aid. Once you know one side and one angle, any missing side follows directly from these ratios.
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