Right Triangle Calculator
Calculate sides, acute angles (degrees and radians with π-fractions), area, perimeter, altitude, inradius, and trigonometric ratios for any right triangle given any 2 parameters.
Right Triangle Calculator
Please provide 2 values below to calculate the other values of a right triangle. If radians are selected as the angle unit, it can take values such as pi/3, pi/4, etc.
A right triangle has one angle equal to 90° (γ = 90° or π/2 rad). To solve all properties, enter any 2 independent values (such as two sides, one side and one angle, or a combination of side/angle with area, perimeter, or altitude). The calculator will automatically identify the geometric theorem and calculate the remaining parameters.
Sides, Area & PerimeterColumn 1
Angles & AltitudeColumn 2
Right Triangle Wireframe
Definition of a Right Triangle
A right triangle (also termed a right-angled triangle or orthogonal triangle) is a polygon with three sides and three interior angles where exactly one angle measures 90 degrees (a right angle, equal to π/2 radians).
In Euclidean plane geometry, the side situated directly opposite the right angle is called the hypotenuse (conventionally denoted as c), which is always the longest side of the triangle. The remaining two perpendicular sides that form the 90° angle are called the legs or catheti (denoted as a and b). The two non-right angles (α and β) are acute angles and are strictly complementary (α + β = 90°).
Core Trigonometric Ratios (SOH CAH TOA)
The fundamental trigonometric functions are defined as the ratios of the lengths of the sides of a right triangle relative to an acute angle α:
Reciprocal Trigonometric Functions:
- Cosecant: csc(α) = 1 / sin(α) = c / a
- Secant: sec(α) = 1 / cos(α) = c / b
- Cotangent: cot(α) = 1 / tan(α) = b / a
Right Triangle Formulas Reference
| Property | Primary Formula | Description & Notes |
|---|---|---|
| Pythagorean Theorem | a² + b² = c² | Relates the squares of the two legs to the hypotenuse square. |
| Area (A) | A = ½ · a · b = ½ · c · h | Equal to half the product of the legs, or half the hypotenuse times altitude. |
| Perimeter (P) | P = a + b + c | Sum of all three boundary lengths. |
| Altitude to Hypotenuse (h) | h = (a · b) / c | Perpendicular distance from the 90° vertex C to hypotenuse c. |
| Inradius (r) | r = (a + b - c) / 2 | Radius of the inscribed circle tangent to all three sides. |
| Circumradius (R) | R = c / 2 | Radius of circumscribed circle passing through vertices A, B, and C (Thales theorem). |
| Median to Hypotenuse (m_c) | m_c = c / 2 = R | Length from right-angle vertex C to midpoint of hypotenuse. |
Special Right Triangles & Pythagorean Triples
45°-45°-90° Triangle
Ratio 1 : 1 : √2An isosceles right triangle where both legs are equal in length (a = b). The hypotenuse is exactly c = a√2. The acute angles are both 45° (π/4 rad).
30°-60°-90° Triangle
Ratio 1 : √3 : 2Formed by bisecting an equilateral triangle. The shorter leg opposite 30° is x, the longer leg opposite 60° is x√3, and the hypotenuse is 2x.
Common Integer Pythagorean Triples
A Pythagorean triple consists of three positive integers (a, b, c) satisfying a² + b² = c². The most frequent primitive triples include:
Practical Applications of Right Triangles
🏗️ Construction & Carpentry (3-4-5 Rule)
Builders use the 3-4-5 rule to verify square 90-degree corners when laying building foundations, decks, walls, and framing.
🗺️ Land Surveying & Triangulation
Surveyors measure distances to inaccessible objects and determine elevation changes across terrain using theodolite angles and right-triangle trigonometry.
🧭 Navigation & Aviation
Pilots and navigators calculate ground speed, crosswind drift angles, bearing intercepts, and true heading using vector resolution into orthogonal components.
🎮 Computer Graphics & Game Physics
3D game engines, ray tracing shaders, and collision detection algorithms rely on right-triangle dot products, trigonometric projections, and Euclidean distances.
Frequently Asked Questions
How many values do I need to solve a right triangle?
Because one angle is already fixed at 90 degrees (γ = 90°), you only need 2 additional independent values (with at least one linear measurement such as a side, perimeter, altitude, or area) to compute all remaining sides, acute angles, and geometric properties.
Can I solve a right triangle if I only know the two acute angles?
No. Entering only angles determines the shape of the triangle (AAA similarity) but not its physical scale or size. At least one linear side, perimeter, altitude, or area is required to fix its exact dimensions.
How do I input angles in radians like pi/3 or pi/4?
Select "radian" from the unit dropdown next to angle α or β, and type mathematical expressions directly into the field, such as pi/3, pi/4, 2*pi/5, or decimal values like 0.5236.
What is the altitude to the hypotenuse formula?
By equating two expressions for the area of the triangle (Area = ½ · a · b = ½ · c · h), the altitude h is given by h = (a · b) / c.
Understanding Right Triangle Calculator & Formula
The Right Triangle Calculator on CalculatorScale.com is engineered to deliver instant, mathematically verified calculations conforming strictly to standard institutional equations. Whether planning long-term investments, managing household budgets, tracking clinical fitness targets, analyzing academic genetics, calculating veterinary dosages, or converting dimensional units, every calculation updates dynamically in real time without lag.
Every formula used in this tool adheres to certified international standards, including World Health Organization biometric metrics, veterinary clinical guidelines, NIST physical conversion constants, and standard biological equations. Results are computed entirely client-side inside your browser with complete data privacy, ensuring your sensitive numbers and statistics never leave your device.
Our calculators are 100% free with no registration, subscriptions, or paywalls required. You can toggle between imperial and metric units, customize input parameters, export calculation summaries, or embed this calculator directly into your own website or blog.
How to Use the Right Triangle Calculator
- 1 Enter any 2 known values (e.g. Leg a = 3, Hypotenuse c = 5, or an angle and a side).
- 2 Select degree or radian for angle inputs. Radian inputs accept mathematical expressions like pi/3 or 0.5236.
- 3 Click the green "Calculate ▶️" button to solve all triangle properties.
- 4 Review the scaled geometric diagram, comprehensive metrics table, SOH CAH TOA trigonometric values, and step-by-step derivation.
Right Triangle Calculator — Frequently Asked Questions
Q1 How many inputs are required to solve a right triangle? ▾
Since a right triangle always has one 90-degree angle (γ = 90° or π/2 rad), exactly 2 additional independent parameters (with at least one linear dimension such as a leg, hypotenuse, perimeter, altitude, or area) are required to solve all remaining dimensions.
Q2 Why can I not calculate the triangle with only two angles? ▾
Two angles determine the angular shape of the triangle (AAA similarity) but do not establish its physical scale. At least one linear measurement is needed to fix its absolute size.
Q3 How do I input radians like pi/3 or pi/4? ▾
Select "radian" from the angle dropdown and enter pi/3, pi/4, 2*pi/5, or decimals. The solver automatically evaluates π fractions and expressions.
Q4 What is the altitude to hypotenuse formula? ▾
By equating the two formulas for triangle area: Area = ½ × a × b = ½ × c × h, we get h = (a × b) / c.
More in Triangle calculators
Pythagorean Theorem Calculator
Right TrianglesSolve right triangle side lengths (a² + b² = c²), acute angles, altitude, area, and perimeter with exact radical support (k√r).
30 60 90 Triangle Calculator
TrianglesSolve 30°-60°-90° special right triangle: side ratios 1 : √3 : 2.
45 45 90 Triangle Calculator
TrianglesSolve 45°-45°-90° isosceles right triangle: side ratios 1 : 1 : √2.
Area of a Right Triangle Calculator
TrianglesCalculate area of right triangle A = 0.5 × a × b from legs a and b.
Circumscribed Circle Calculator
TrianglesCalculate circumradius R = (a b c) / (4 Area) of triangle circumcircle.
Classifying Triangles Calculator
TrianglesClassify triangle by side lengths (equilateral, isosceles, scalene) and angles (acute, right, obtuse).
Explore All 4,700+ Calculators by Category
Genetics, molecular laboratory science, veterinary dosages, animal gestation, agronomy, and forestry.
Stoichiometry, chemical solutions, thermodynamics, kinetics, electrochemistry, and organic reactions.
Concrete, roofing, brick, framing, lumber spans, driveway paving, gravel, and material specifications.
Precision metric and imperial unit converters for length, area, volume, mass, pressure, temperature, and speed.
Carbon footprint, solar energy payback, tree sequestration, recycling offsets, and sustainable living metrics.
Travel durations, clothing and shoe sizes, fuel economy, home budgeting, sleep cycles, and daily routines.
Mortgage amortization, compound interest, loans, retirement planning, taxes, investments, and salary projections.
Culinary volume-to-weight conversions, baking hydration, ingredient scaling, pizza dough, and party food planning.
BMI, daily caloric needs (TDEE/BMR), body fat percentage, pregnancy due dates, clinical dosage, and pediatric metrics.
Scientific calculators, fractions, percentages, geometry, trigonometry, algebra, linear systems, and matrices.
Kinematics, Newtonian dynamics, thermodynamics, fluid mechanics, electromagnetism, optics, and relativity.
Heart rate training zones, running pace, triathlon pacing, cycling power, calories burned, and 1RM lifting benchmarks.
Standard deviation, normal distribution (Z-scores), confidence intervals, sample size, and probability odds.
Education GPA, photography exposures, music BPM, data storage, network bandwidth, and specialty calculations.