Руководство по формулам геометрииa = √(c² - b²)

Как найти неизвестный катет a или b прямоугольного треугольника

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.

Instant Interactive Widget5 Visual Worked ExamplesExact Radical & Decimal Outputs

Как найти катет a прямоугольного треугольника (формула и разбор)

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:

Solving for Leg a

a = √(c² - b²)

Subtract base leg b² from hypotenuse c² when c and b are known. Always verify that c > b first.

Solving for Leg b

b = √(c² - a²)

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.

Common Mistake Check: Never Add When Solving for a LegIf you mistakenly compute 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.

Enter hypotenuse c and one known leg to solve immediately in exact radical and decimal form.

Hypotenuse (c)Longest side (c > b)

Known Leg b0 < b < c

Missing Leg a Solution

a = 12

1. Formula: a = √(c² - b²)

2. Substitute: a = √(13² - 5²)

3. Subtract: a = √(169 - 25) = √144

4. Result: a = 12

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Пропорциональный SVG прямоугольного треугольника

Interactive Pythagorean theorem formula diagram illustrating perpendicular legs and right triangle hypotenuse proportions.

Прямоугольный треугольник (90°) • Площадь (A): 30²

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