Right Triangle Side and Angle Calculator

With two known values, use the Pythagorean theorem or trigonometry. If you know both legs (a, b): hypotenuse c = sqrt(a squared plus b squared). If you know one leg and hypotenuse: missing leg = sqrt(c squared minus known leg squared). If you know an angle and one side: use sin, cos, or tan to find the others. This calculator handles all four cases automatically.

Use inverse trigonometric functions. Given legs a and b: angle A = arctan(a/b) in degrees. Given one leg a and hypotenuse c: angle A = arcsin(a/c). Given leg b and hypotenuse c: angle A = arccos(b/c). The right angle C is always 90 degrees, and angle B = 90 minus angle A.

The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: c squared = a squared + b squared, where c is the hypotenuse. Equivalently, c = sqrt(a squared + b squared). For example, legs 3 and 4 give hypotenuse = sqrt(9 + 16) = sqrt(25) = 5. The 3-4-5 triple is the most famous Pythagorean triple.

SOH, CAH, TOA is a mnemonic for the three primary trig ratios in a right triangle. SOH: Sin(A) = Opposite divided by Hypotenuse. CAH: Cos(A) = Adjacent divided by Hypotenuse. TOA: Tan(A) = Opposite divided by Adjacent. These let you find any missing side when an angle is known: opposite = hypotenuse times sin(A), adjacent = hypotenuse times cos(A), opposite = adjacent times tan(A).

The six trig ratios for angle A are: sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent, csc A = hypotenuse/opposite (reciprocal of sin), sec A = hypotenuse/adjacent (reciprocal of cos), cot A = adjacent/opposite (reciprocal of tan). This calculator shows all six for both acute angles once the triangle is solved.

Area = one half times leg a times leg b. Since the two legs are perpendicular, one acts as the base and the other as the height: Area = 0.5 times a times b. For the 3-4-5 right triangle: Area = 0.5 times 3 times 4 = 6 square units. Once you know both legs (which this calculator computes in all modes), the area is always computable.

The altitude from the right angle perpendicular to the hypotenuse has length h = (a times b) divided by c, where a and b are the legs and c is the hypotenuse. It is also the geometric mean: h squared = (projection of a onto c) times (projection of b onto c). This altitude divides the triangle into two smaller triangles that are each similar to the original.

The two key special right triangles are: the 45-45-90 triangle (isosceles right triangle, sides 1:1:sqrt 2) and the 30-60-90 triangle (sides 1:sqrt 3:2). They appear constantly in geometry, trigonometry, and construction because their trig ratios are exact values: sin 30 = 0.5, sin 45 = sqrt 2 over 2, sin 60 = sqrt 3 over 2. These are the exact values tested on standardized exams and used in engineering.

Use sine for the opposite side and cosine for the adjacent side. Opposite = hypotenuse times sin(angle). Adjacent = hypotenuse times cos(angle). Example: angle A = 37 degrees, hypotenuse = 10. Opposite leg a = 10 times sin(37) = 10 times 0.6018 = 6.02. Adjacent leg b = 10 times cos(37) = 10 times 0.7986 = 7.99. Then check: sqrt(6.02 squared + 7.99 squared) should equal 10.

If you know the angle and the opposite leg: hypotenuse = opposite divided by sin(angle). If you know the angle and the adjacent leg: hypotenuse = adjacent divided by cos(angle). Example: angle A = 30 degrees, opposite leg a = 5. Hypotenuse = 5 divided by sin(30) = 5 divided by 0.5 = 10. Adjacent leg b = sqrt(100 minus 25) = sqrt(75) = 8.66.

Right triangles appear everywhere: architects use them to check that corners are square (the 3-4-5 rule). Surveyors use trigonometry to measure land without direct access. Navigation uses bearing angles and distances forming right triangles. Ramps, stairs, and roof pitches are analyzed as right triangles. GPS and satellite positioning uses triangulation. Electricians use them to find wire lengths when routing at angles through walls.

This calculator is specialized for right triangles, which always have one 90-degree angle. Because one angle is fixed, you only need two pieces of information (instead of three for a general triangle) to fully solve the triangle. The calculator uses the Pythagorean theorem and basic trig ratios, which are simpler and more exact than the Law of Sines and Law of Cosines used for general triangles. It also computes all six trig ratios and the altitude to the hypotenuse.