Solve any right triangle — find hypotenuse, legs, angles, area and perimeter using the Pythagorean theorem and trigonometric functions.
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The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the two legs: a² + b² = c². So c = √(a² + b²). For a triangle with legs 3 and 4: c = √(9 + 16) = √25 = 5. This applies only to triangles with one 90° angle.
Use the Pythagorean theorem: c = √(a² + b²). For example, with legs a = 6 and b = 8: c = √(36 + 64) = √100 = 10. If you know the hypotenuse and one leg: a = √(c² − b²). Check your answer by verifying: a² + b² should equal c² exactly.
Use inverse trigonometric functions. If you know legs a and b: Angle A = arctan(a/b) and Angle B = arctan(b/a). These two angles always add to 90°. For legs 3 and 4: Angle A = arctan(3/4) = 36.87° and Angle B = arctan(4/3) = 53.13°. The right angle is always 90°, so all three angles sum to 180°.
Pythagorean triples are whole-number sets that satisfy a² + b² = c². Common ones: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Multiples also work: 6-8-10, 9-12-15 (multiples of 3-4-5). These are extremely useful in construction — a 3-4-5 right triangle lets you check for square corners without any trigonometry.
The 3-4-5 rule is a practical application of the Pythagorean theorem for squaring corners. Measure 3 units along one wall, 4 units along the other wall, and check that the diagonal between the two endpoints measures exactly 5 units. If it does, the corner is a perfect 90°. Any multiple works: 6-8-10, 9-12-15, 30-40-50.
Area = (1/2) × base × height. For a right triangle, the two legs are the base and height: Area = 0.5 × a × b. For a right triangle with legs 6 and 8: Area = 0.5 × 6 × 8 = 24 square units. Alternatively, Area = (1/2) × c × h, where c is the hypotenuse and h is the altitude from the right angle to the hypotenuse.
SOH-CAH-TOA is the mnemonic: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent — all relative to a given angle. For angle A with opposite side a, adjacent side b, hypotenuse c: sin(A) = a/c, cos(A) = b/c, tan(A) = a/b. These ratios remain constant for any triangle with the same angles.
A 45-45-90 triangle has legs of equal length (a = b) and a hypotenuse of a√2. If each leg is 1, the hypotenuse is √2 ≈ 1.414. This appears when you cut a square diagonally. If you know one leg: hypotenuse = leg × √2. If you know the hypotenuse: leg = hypotenuse / √2 = hypotenuse × √2/2.
A 30-60-90 triangle has sides in the ratio 1 : √3 : 2. The shortest side (opposite 30°) = half the hypotenuse. The medium side (opposite 60°) = (√3/2) × hypotenuse ≈ 0.866 × hypotenuse. For hypotenuse = 10: short side = 5, medium side = 5√3 ≈ 8.66. These triangles appear when you bisect an equilateral triangle.
Right triangles are fundamental in construction: checking square corners (3-4-5 rule), calculating rafter lengths from roof span and pitch, determining stair rise and run, finding ladder placement distances, calculating the length of diagonal bracing, and measuring inaccessible distances indirectly. Any slope, angle, or diagonal measurement involves right triangle trigonometry.
