Right Triangle Calculator Online Free
Right Triangle Calculator
Triangle Properties
Right triangle with sides a, b, c and angles α, β, and 90°
Instructions: Enter any 2 values (sides or angles) and the calculator will automatically compute all other properties. For radians, you can use expressions like π/3, π/4, π/6, etc.
Understanding Right Triangles - Complete Guide
A right triangle (also called a right-angled triangle) is a triangle in which one angle is exactly 90 degrees (a right angle). This special property makes right triangles fundamental to geometry, trigonometry, and countless real-world applications. The side opposite the right angle is called the hypotenuse and is always the longest side of the triangle.
Right triangles are governed by the Pythagorean theorem and form the foundation of trigonometric functions. Understanding right triangles is essential for fields ranging from architecture and engineering to navigation and computer graphics.
Essential Formulas
Pythagorean Theorem
Where c is the hypotenuse and a, b are the legs
Area Formula
Area equals half the product of the two legs
Height to Hypotenuse
Altitude dropped from right angle to hypotenuse
Perimeter Formula
Sum of all three sides
Trigonometric Relationships
Right triangles are the foundation of trigonometry. The six trigonometric functions are defined based on the ratios of sides in a right triangle:
Sine (sin)
Opposite / Hypotenuse
Cosine (cos)
Adjacent / Hypotenuse
Tangent (tan)
Opposite / Adjacent
Cosecant (csc)
Hypotenuse / Opposite
Secant (sec)
Hypotenuse / Adjacent
Cotangent (cot)
Adjacent / Opposite
Remember: In a right triangle, α + β = 90° (or π/2 radians). The two acute angles are always complementary.
Worked Examples
1Finding All Properties from Two Sides
Problem: A right triangle has legs a = 3 and b = 4. Find all other properties.
Step 1: Find hypotenuse using Pythagorean theorem
c² = a² + b² = 3² + 4² = 9 + 16 = 25
c = √25 = 5
Step 2: Calculate angles
α = arctan(a/b) = arctan(3/4) ≈ 36.87°
β = arctan(b/a) = arctan(4/3) ≈ 53.13°
Step 3: Calculate height to hypotenuse
h = (a × b) / c = (3 × 4) / 5 = 2.4
Step 4: Calculate area and perimeter
A = (a × b) / 2 = (3 × 4) / 2 = 6
P = a + b + c = 3 + 4 + 5 = 12
✓ Answer: c = 5, α ≈ 36.87°, β ≈ 53.13°, h = 2.4, A = 6, P = 12
2Using One Side and One Angle
Problem: A right triangle has side a = 5 and angle α = 30°. Find all other properties.
Step 1: Convert angle to radians (if needed): 30° = π/6 ≈ 0.5236 rad
Step 2: Find side b using tangent
tan(α) = a / b
b = a / tan(α) = 5 / tan(30°) = 5 / 0.5774 ≈ 8.66
Step 3: Find hypotenuse using sine
sin(α) = a / c
c = a / sin(α) = 5 / sin(30°) = 5 / 0.5 = 10
Step 4: Calculate other angle
β = 90° - α = 90° - 30° = 60°
Step 5: Calculate remaining properties
h = (5 × 8.66) / 10 ≈ 4.33
A = (5 × 8.66) / 2 ≈ 21.65
P = 5 + 8.66 + 10 = 23.66
✓ Answer: b ≈ 8.66, c = 10, β = 60°, h ≈ 4.33, A ≈ 21.65, P ≈ 23.66
3Hypotenuse and One Angle
Problem: A right triangle has hypotenuse c = 13 and angle β = 45°. Find all properties.
Step 1: Calculate complementary angle
α = 90° - β = 90° - 45° = 45°
Note: This is an isosceles right triangle (45-45-90 triangle)
Step 2: Find side a using sine
sin(α) = a / c
a = c × sin(α) = 13 × sin(45°) = 13 × 0.7071 ≈ 9.19
Step 3: Find side b using cosine
cos(α) = b / c
b = c × cos(α) = 13 × cos(45°) = 13 × 0.7071 ≈ 9.19
Step 4: Calculate remaining properties
h = (9.19 × 9.19) / 13 ≈ 6.50
A = (9.19 × 9.19) / 2 ≈ 42.25
P = 9.19 + 9.19 + 13 = 31.38
✓ Answer: a ≈ 9.19, b ≈ 9.19, α = 45°, h ≈ 6.50, A ≈ 42.25, P ≈ 31.38
Special Right Triangles
Certain right triangles have special angle relationships and side ratios that appear frequently in mathematics and applications:
45°-45°-90° Triangle
Isosceles right triangle
If legs = x, then hypotenuse = x√2
Common in: Square diagonals, octagon construction
30°-60°-90° Triangle
Half of an equilateral triangle
If short leg = x, then long leg = x√3, hypotenuse = 2x
Common in: Hexagon construction, roof design
Common Pythagorean Triples
These are sets of three positive integers (a, b, c) where a² + b² = c²:
Most common
Architecture
Construction
Engineering
Tip: Any multiple of a Pythagorean triple is also a Pythagorean triple (e.g., 6-8-10, 9-12-15).
Real-World Applications
🏗️Architecture & Construction
- Calculating roof pitch and rafter lengths
- Ensuring walls are perpendicular (3-4-5 method)
- Designing staircases and ramps
- Determining building heights using shadows
🧭Navigation & Surveying
- Calculating distances using triangulation
- Determining altitude of aircraft
- GPS coordinate calculations
- Marine and aviation navigation
⚙️Physics & Engineering
- Resolving force vectors into components
- Analyzing inclined planes and ramps
- Calculating projectile motion trajectories
- Mechanical linkage design
🖥️Computer Graphics
- 3D rendering and perspective calculations
- Screen diagonal and aspect ratio determination
- Pixel distance calculations
- Animation and game physics
⚽Sports & Recreation
- Baseball diamond measurements
- Basketball court line calculations
- Golf course distance estimation
- Rock climbing angle calculations
🔭Astronomy
- Measuring celestial object heights above horizon
- Calculating distances to stars using parallax
- Telescope angle adjustments
- Orbital mechanics calculations
Tips & Best Practices
Historical Context
Right triangles have been studied since ancient times. The Pythagorean theorem, while named after Pythagoras (c. 570-495 BCE), was known to the Babylonians over 1,000 years earlier. Clay tablets from around 1800 BCE show lists of Pythagorean triples, demonstrating sophisticated mathematical knowledge.
Ancient Egyptian builders used the 3-4-5 right triangle to create perfect right angles when constructing pyramids and temples. They would use ropes with 12 equally spaced knots to form a triangle with sides of 3, 4, and 5 units, ensuring a 90-degree angle.
The development of trigonometry by ancient Greek, Indian, and Islamic mathematicians was built entirely on the properties of right triangles. Today, right triangle calculations are fundamental to GPS systems, computer graphics, architectural design, and countless other modern technologies.
Advanced Topics
Similar Right Triangles
When a right triangle's altitude is drawn to the hypotenuse, it creates three similar triangles: the original triangle and two smaller triangles. This property is used in geometric mean calculations and scaling problems.
Inverse Trigonometric Functions
Functions like arcsin, arccos, and arctan (also written as sin⁻¹, cos⁻¹, tan⁻¹) are used to find angles when you know the side ratios. These are essential for solving right triangles when sides are known but angles are not.
3D Applications
Right triangles extend into three dimensions for calculating spatial diagonals, vector projections, and distances in 3D space. The distance formula in 3D (d = √(x² + y² + z²)) is a direct extension of the Pythagorean theorem.
Quick Reference Guide
Basic Formulas
- • a² + b² = c²
- • A = (a × b) / 2
- • P = a + b + c
- • h = (a × b) / c
Angle Relationships
- • α + β = 90°
- • sin(α) = cos(β)
- • tan(α) = 1 / tan(β)
- • sin²(α) + cos²(α) = 1
Special Triangles
- • 45-45-90: 1 : 1 : √2
- • 30-60-90: 1 : √3 : 2
- • 3-4-5 (Pythagorean triple)
Conversions
- • Degrees to radians: × (π/180)
- • Radians to degrees: × (180/π)
- • π ≈ 3.14159
Right triangles are among the most important geometric shapes in mathematics and applied sciences. Whether you're designing a building, programming a video game, navigating with GPS, or studying physics, understanding right triangles and their properties is essential. This calculator provides all the tools you need to work with right triangles efficiently, supporting both degrees and radians for maximum flexibility in academic and professional applications.