Find the hypotenuse, a missing leg, or verify a right triangle — with step-by-step solutions, live triangle visual, angles, area, perimeter, and Pythagorean triple detection.
📐
Find the Hypotenuse (c)
Enter legs a and b to find hypotenuse c
First leg of right triangle
Second leg of right triangle
Real-world:
Enter hypotenuse c and one leg — leave the unknown blank
Known leg (if known)
Known leg (if known)
Longest side (opposite 90°)
Examples:
Enter all 3 sides — the calculator checks if it forms a right triangle
Try:
📐 Step-by-Step Solution
📐 Right Triangle Visual
RESULT
—
—
Side a—
Side b—
Hypotenuse c—
Angle α (at A)—
Angle β (at B)—
Area—
Perimeter—
Altitude (h)—
🌟 Pythagorean Triple!
Pythagorean Theorem Formula & How to Use It
The Pythagorean Theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs:
a² + b² = c²
Find c: c = √(a² + b²)
Find a: a = √(c² − b²)
Find b: b = √(c² − a²)
Common Pythagorean Triples
Pythagorean triples are whole number sets where a²+b²=c². These are commonly used in construction, engineering, and math:
3 · 4 · 5
5 · 12 · 13
8 · 15 · 17
7 · 24 · 25
6 · 8 · 10
9 · 12 · 15
20 · 21 · 29
9 · 40 · 41
12 · 16 · 20
Real-World Uses of the Pythagorean Theorem
🏗️ Construction: Check if corners are perfectly square using the 3-4-5 rule
📺 Screen size: Find the diagonal of a TV or monitor from width and height
🪜 Ladder safety: How long a ladder is needed to reach a certain height
🗺️ Navigation: Find direct distance between two points from east-west and north-south travel
⚽ Sports: Find diagonal distance across a field
🎮 Game dev: Collision detection and character movement distance
Angles of a Right Triangle
Once you know all three sides, you can find the angles: α = arctan(a/b) and β = arctan(b/a). The three angles always sum to 180°: α + β + 90° = 180°.
Converse of the Pythagorean Theorem
The converse theorem lets you verify if a triangle is a right triangle. If a²+b²=c² (where c is the longest side), then it IS a right triangle. This calculator's Mode 3 does this automatically. It also tells you if the triangle is acute (a²+b² > c²) or obtuse (a²+b² < c²).
Special Right Triangles
45-45-90 triangle: Legs are equal (a=b). Hypotenuse = a√2. Example: a=b=1 → c=√2≈1.414
30-60-90 triangle: If short leg=a, then long leg=a√3 and hypotenuse=2a
3-4-5 triangle: The most famous Pythagorean triple — used in construction to check right angles
How to Find Triangle Altitude on Hypotenuse
Altitude (h) = (a × b) ÷ c
Example: a=3, b=4, c=5 → h = (3×4)÷5 = 12÷5 = 2.4
Pythagorean Theorem in Real Life
The 3-4-5 rule is used by builders worldwide to check right angles: measure 3 units along one wall, 4 units along the other, and the diagonal should be exactly 5 units. If it is, the corner is a perfect 90°.
The Pythagorean Theorem states: a² + b² = c², where a and b are the two legs and c is the hypotenuse (longest side). It only applies to right triangles. Named after Greek mathematician Pythagoras (~570-495 BC), though the relationship was known in ancient Babylon and India centuries earlier.
c = √(a² + b²). Example: a=3, b=4 → c = √(9+16) = √25 = 5. The hypotenuse is always the longest side and is always opposite the 90° angle. It cannot be longer than the sum of the two legs.
a = √(c² − b²). Example: hypotenuse c=13, leg b=12 → a = √(169−144) = √25 = 5. Always subtract the known leg squared FROM the hypotenuse squared — not the other way around.
Pythagorean triples are sets of three whole numbers (a,b,c) where a²+b²=c². The most famous is (3,4,5). Multiples also work: (6,8,10) is 2×(3,4,5). Builders use the 3-4-5 rule to check right angles — measure 3 units one way, 4 units the other, and the diagonal should be exactly 5 units.
Check if a² + b² = c² (where c is the longest side). If equal → right triangle. If a²+b² > c² → acute triangle (all angles less than 90°). If a²+b² < c² → obtuse triangle (one angle greater than 90°). Use Mode 3 of this calculator to verify automatically.
A 45-45-90 triangle is a special right triangle where both legs are equal and the two acute angles are both 45°. The hypotenuse = leg × √2. Example: if both legs = 5, hypotenuse = 5√2 ≈ 7.071. This is also called an isosceles right triangle.
Construction: The 3-4-5 rule checks if corners are exactly 90°. Screen size: diagonal of a TV from width and height. Navigation: direct distance between two GPS points. Engineering: structural calculations. Game development: distance and collision detection. Architecture: roof pitch calculations.
CalcVelo Pythagorean Theorem Calculator — Free online tool to find hypotenuse (a²+b²=c²), missing leg, or verify right triangle. Includes angles, area, perimeter, altitude, live visual, and Pythagorean triple detection.
