Jak Obliczyć Brakującą Przyprostokątną a lub b
When you already know the hypotenuse c and one side of a right triangle, Pythagorean theorem solving for a or b shifts from addition to subtraction: a = √(c² - b²). Need to solve angles, area, or 3D space diagonals too? Open our full Pythagoras equation solver on the homepage.
Jak Wyznaczyć Bok a (Wzór i Dowód)
The standard Pythagorean identity a² + b² = c² isolates either perpendicular leg simply by subtracting the other squared leg from c² and taking the positive square root:
Subtract base leg b² from hypotenuse c² when c and b are known. Always verify that c > b first.
Because legs a and b are symmetric perpendicular sides, solving for leg b uses the exact same subtraction identity when c and a are known.
c² + b² instead of c² - b², your missing leg will come out longer than the hypotenuse — which is geometrically impossible in Euclidean right triangles where c > a and c > b.Inline Missing Leg Calculator (a = √(c² - b²))
Enter hypotenuse c and one known leg to solve immediately in exact radical and decimal form.
5 Worked Examples: Solving for Missing Leg a or b
Walk through integers, word problems, simplified radicals, prime square roots, and decimals step by step.
Example 1: Classic Integer Triple (Find Leg a)
Given hypotenuse c = 5 and base leg b = 4, find height leg a.
- 1.Start with rearranged formula: a = √(c² - b²)
- 2.Substitute c = 5 and b = 4: a = √(5² - 4²)
- 3.Square and subtract: a = √(25 - 16) = √9
- 4.Extract exact root: a = 3
Example 2: Wall & Ladder Word Problem (Solve for a)
A 13 ft ladder leans against a wall with its base 5 ft away. Find wall height a.
- 1.Identify longest side c = 13 ft and base leg b = 5 ft
- 2.Apply subtraction rule: a = √(13² - 5²)
- 3.Compute squares and difference: a = √(169 - 25) = √144
- 4.Simplify perfect square: a = 12 ft
Example 3: Simplifying a Factored Radical Result
Find leg b when hypotenuse c = 9 and leg a = 6 in exact simplified radical form.
- 1.Rearrange equation for missing leg b: b = √(c² - a²)
- 2.Substitute c = 9 and a = 6: b = √(9² - 6²)
- 3.Subtract squares: b = √(81 - 36) = √45
- 4.Extract perfect square factor 9: b = √(9 × 5) = 3√5 ≈ 6.7082
Example 4: Prime Irreducible Radical Answer
Solve for leg a when hypotenuse c = 4 and leg b = 3.
- 1.Write the formula for leg a: a = √(c² - b²)
- 2.Substitute values: a = √(4² - 3²)
- 3.Compute difference of squares: a = √(16 - 9) = √7
- 4.7 is prime (no square factor > 1), so exact leg a = √7 ≈ 2.6458
Example 5: Decimal Construction Measurement
A sloped roof truss has rafter hypotenuse c = 10.5 m and half-span b = 6.3 m. Find rise a.
- 1.Set up Pythagorean subtraction equation: a = √(c² - b²)
- 2.Substitute decimal measurements: a = √(10.5² - 6.3²)
- 3.Subtract squares: a = √(110.25 - 39.69) = √70.56
- 4.Evaluate square root: a = 8.4 m