Find all measurements of a right triangle from two known sides. Calculate area, perimeter, angles, and more.
Last updated: April 2026 | By Patchworkr Team
A right triangle is a triangle with one angle measuring exactly 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse (the longest side), and the other two sides are called legs. Right triangles are fundamental in geometry, trigonometry, and physics because they have unique mathematical properties.
The most important property is the Pythagorean theorem: a² + b² = c², where a and b are the legs and c is the hypotenuse. This relationship allows you to find any missing side if you know the other two. Right triangles appear everywhere in construction, surveying, navigation, and engineering. The 3-4-5 triangle is the most famous example and is commonly used in construction to create square corners.
Right triangles also relate directly to trigonometric functions (sine, cosine, tangent), making them essential for calculating angles and distances. Whether you're building a roof, measuring heights, or solving physics problems, understanding right triangles is crucial for practical problem-solving.
You must have at least two of the three sides. You can provide either: (a) both legs (two perpendicular sides), or (b) one leg and the hypotenuse (longest side).
Why: The Pythagorean theorem requires two known values to solve for the third. Having at least two measurements ensures a unique solution. This determines which calculation mode to use.
If you have both legs, calculate: c = √(a² + b²). If you have one leg and hypotenuse, calculate the missing leg: missing = √(c² - known²).
Why: The Pythagorean theorem is the fundamental relationship in right triangles. It only applies to right triangles and provides exact results without approximation.
Once all three sides are known, these calculations are straightforward. Area represents internal space; perimeter represents the distance around the triangle's edges.
Why: Area and perimeter are practical measurements used in real-world applications: area for material coverage, perimeter for edging or framing materials.
Use inverse trigonometric functions (arcsin, arccos, arctan) to convert ratios into actual angle measurements in degrees or radians.
Why: Angles determine the triangle's shape and are essential for many engineering, construction, and physics applications. Knowing all angles provides complete geometric information.
Check that: the hypotenuse is longer than both legs, angles sum to 180°, and the Pythagorean theorem holds. Verify numerical values are reasonable for the real-world context.
Why: Verification catches computational errors and ensures the triangle's properties are mathematically consistent. Interpretation confirms results make practical sense for their intended application.
Roof Rafter Calculation
For any right triangle: a² + b² = c², where a and b are legs and c is the hypotenuse. It's one of the most fundamental theorems in geometry.
The 3-4-5 triangle. It's the simplest Pythagorean triple and is used in construction to verify right angles.
No. Check: 5² + 10² = 25 + 100 = 125 ≠ 15² = 225. These sides don't form a valid right triangle.
The circumradius (radius of circle through all vertices) equals half the hypotenuse. For a 3-4-5 triangle, it's 2.5.
Use inverse trigonometric functions: angle = arctan(opposite/adjacent) or arcsin(opposite/hypotenuse).
The inradius is the radius of the circle inscribed inside the triangle. For right triangles: r = (a + b - c) / 2.
Yes! If the two legs are equal, it's an isosceles right triangle with 45-45-90 angles. Example: 1-1-√2.
The perpendicular line from the right angle to the hypotenuse. Its length = (leg₁ × leg₂) ÷ hypotenuse.
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