Calculate Right Triangle

Pythagoras and right angles: Complete calculation from two known sides

Right Triangle Calculator

Pythagorean Theorem

A right triangle has a 90° angle. The Pythagorean theorem enables calculation of all parameters from two known side lengths.

Select Two Known Parameters
Value for the first parameter
Value for the second parameter
Calculated Properties
Leg a:
Leg b:
Hypotenuse c:
Height h:
Area A:
Perimeter P:
Segment p:
Segment q:
Angle α:
Angle β:

Right Triangle

Right triangle

Right triangle with legs a, b and hypotenuse c.
The right angle (90°) lies between the two legs.

What is a Right Triangle?

A right triangle is a triangle with exactly one 90° angle:

  • Right angle: Exactly one angle measures 90° (right angle)
  • Legs: The two sides that form the right angle
  • Hypotenuse: The longest side, opposite the right angle
  • Pythagoras: a² + b² = c² (fundamental theorem)
  • Applications: Construction, navigation, physics, technology
  • Special case: All other triangle types as generalization

The Pythagorean Theorem

The Pythagorean theorem is one of the most famous theorems in mathematics:

Pythagorean Formula
\[a^2 + b^2 = c^2\]

Squares of legs = Square of hypotenuse

Historical Significance
  • Known since ca. 2000 BC
  • Named after Pythagoras (~ 570-495 BC)
  • Foundation of Euclidean geometry
  • Over 400 different proofs known

Practical Applications

Right triangles and Pythagoras are everywhere in practice:

Construction & Architecture
  • Roof constructions and gables
  • Stairs and ramps
  • Right angle verification for foundations
  • Structural calculations
Navigation & Technology
  • GPS and triangulation
  • Distance measurement
  • Computer graphics and 3D models
  • Physics: force decomposition

Calculation Methods

Right triangles can be calculated in various ways:

Pythagoras

Calculate the third side from two side lengths

Trigonometry

With sine, cosine, and tangent

Altitude theorem

Via height and hypotenuse segments

Formulas for Right Triangles

Pythagorean Theorem
\[c = \sqrt{a^2 + b^2}\]

Hypotenuse from both legs

Leg Formulas
\[a = \sqrt{c^2 - b^2}\]

Leg from hypotenuse and other leg

Area
\[A = \frac{a \cdot b}{2}\]

Half the product of the legs

Height to Hypotenuse
\[h = \frac{a \cdot b}{c}\]

Product of legs divided by hypotenuse

Trigonometric Formulas
Angle α:
\[\alpha = \arcsin\left(\frac{a}{c}\right) = \arccos\left(\frac{b}{c}\right)\]
Angle β:
\[\beta = \arcsin\left(\frac{b}{c}\right) = \arccos\left(\frac{a}{c}\right)\]
Hypotenuse Segments
Segment p: \(p = \frac{a^2}{c}\)
Segment q: \(q = \frac{b^2}{c}\)
Altitude theorem: \(h = \sqrt{p \cdot q}\)
Leg theorem: \(a^2 = p \cdot c\)
Notation and Symbols
  • a, b: Legs (adjacent to right angle)
  • c: Hypotenuse (longest side)
  • h: Height to hypotenuse
  • A: Area
  • P: Perimeter (P = a + b + c)
  • p, q: Hypotenuse segments
  • α, β: Acute angles (α + β = 90°)
  • γ: Right angle (90°)

Calculation Example

Given
Leg a = 4 Leg b = 8

Two legs known

1. Calculate Hypotenuse
\[c = \sqrt{4^2 + 8^2} = \sqrt{16 + 64} = \sqrt{80} \approx 8.94\]

Apply Pythagorean theorem

2. Calculate Area
\[A = \frac{4 \times 8}{2} = 16\]

Half the product of the legs

3. Height to Hypotenuse
\[h = \frac{4 \times 8}{8.94} \approx 3.58\]

Legs product divided by hypotenuse

4. Calculate Angles
\[\alpha = \arctan\left(\frac{4}{8}\right) = \arctan(0.5) \approx 26.57°\] \[\beta = 90° - 26.57° = 63.43°\]

With tangent and angle relationship

Right Triangles in Mathematics and Practice

Right triangles are not just geometric objects, but fundamental building blocks of mathematics and practical applications. The Pythagorean theorem, which describes their properties, is among the most famous and important theorems in mathematics.

The Right Angle - Definition and Significance

A right triangle is uniquely characterized by the 90° angle:

  • Right angle: Exactly 90° = π/2 radians = quarter of a full circle
  • Legs: The two sides that form the right angle
  • Hypotenuse: The side opposite the right angle (longest side)
  • Acute angles: The two other angles are acute and sum to 90°
  • Special position: Right triangles are special cases of general triangles

The Pythagorean Theorem - Historical Context

The Pythagorean theorem is one of the oldest and most frequently proven mathematical results:

Historical Development

Babylonian and Egyptian mathematicians already knew special right triangles (e.g., 3-4-5) around 2000 BC.

Pythagoras of Samos

The Greek philosopher and mathematician (ca. 570-495 BC) is associated with the first general proof.

Universal Significance

The theorem was discovered independently in various cultures and forms the foundation of Euclidean geometry.

Modern Relevance

Fundamental importance in analysis, physics, computer science, and all technical disciplines.

Practical Applications in the Modern World

Right triangles are present everywhere in our daily lives:

Construction and Architecture
  • Roof constructions and gables
  • Stair and ramp inclinations
  • Right angle verification for foundations
  • Structural calculations of supporting structures
Navigation and Surveying
  • GPS triangulation
  • Distance measurement without direct access
  • Height measurement with angle measurement
  • Cartographic projections
Technology and Computer Science
  • 3D computer graphics and games
  • Image processing and computer vision
  • Robotics and autonomous systems
  • Signal and image compression
Physics and Natural Sciences
  • Force decomposition in mechanics
  • Wave theory and interference
  • Optics: refraction and reflection
  • Electrical engineering: AC analysis

Advanced Mathematical Concepts

Right triangles are the starting point for many advanced mathematical concepts:

  • Trigonometry: Sine, cosine, and tangent arise from right triangles
  • Analytic geometry: Orthogonality and dot product
  • Complex numbers: Representation in the Gaussian number plane
  • Vector calculus: Orthogonal vectors and projections
  • Linear algebra: Orthogonal matrices and QR decomposition
  • Analysis: Fourier analysis and orthogonal function systems

Special Right Triangles

Famous Number Triples

Pythagorean triples are integer solutions: (3,4,5), (5,12,13), (8,15,17), (7,24,25), etc.

45°-45°-90° Triangle

Isosceles right triangle: legs equal length, ratio 1:1:√2

30°-60°-90° Triangle

Half of an equilateral triangle: side ratio 1:√3:2

Application in Trigonometry

These special triangles provide exact values for trigonometric functions.

Proof Diversity of the Pythagorean Theorem

There are over 400 different proofs for the Pythagorean theorem:

Geometric Proofs

Area comparisons, square constructions, similarity considerations

Algebraic Proofs

Coordinate geometry, vector calculus, complex numbers

Trigonometric Proofs

Via cosine law, sine-square identities

Physical "Proofs"

Energy conservation, mechanical analogies

Generalizations and Extensions

The Pythagorean theorem can be generalized in various directions:

  • Higher dimensions: Euclidean norm in n-dimensional space
  • Non-Euclidean geometries: Modifications for spherical and hyperbolic geometry
  • Law of cosines: Generalization for arbitrary triangles
  • Parallelogram law: Generalization for parallelograms
  • Stewart's theorem: Extension with division lines
Summary

Right triangles and the Pythagorean theorem form a fundamental cornerstone of mathematics. From practical applications in construction to abstract mathematical theories, they permeate all areas of the exact sciences. Their simplicity in definition contrasts with their enormous scope and breadth of application in the modern world.

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Bisector of a triangle  •  Equilateral triangle  •  Right triangles  •  Right triangle, given 1 side and 1 angle  •  Isosceles right triangles  •  Isosceles triangles  •  Triangle area, given 2 sides and 1 anglee  •  Triangle area, given 1 side and 2 angles  •  Triangle, Incircle, given 3 sides  •  Area of a triangle given base and height  •  Triangle vertices, 3 x/y points  •