Right Triangle Calculator – Solve Complete Triangles: Find Sides, Angles, Area & Perimeter

  • Adam
  • November 4, 2025

Free right triangle calculator. Solve from two sides, side & angle, or side & area. Calculate all properties: hypotenuse, angles, area, perimeter with trigonometry. Step-by-step solutions.

Right Triangle Calculator

Solve Complete Right Triangles: Calculate All Sides, Angles, Area, and Perimeter with Detailed Solutions

Quick Navigation

Right Triangle from Two Sides

Enter two sides to calculate all triangle properties including the hypotenuse, angles, area, and perimeter.

Sides & Complete Triangle

Enter two legs (the sides forming the 90° angle) to find hypotenuse and angles

Hypotenuse (c)
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Angle α
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Angle β
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Area
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Perimeter
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Triangle Diagram
a c b 90° α β h
Relationships:
c² = a² + b²
Area = ½ab
P = a + b + c

Right Triangle from Side & Angle

Enter one side and one acute angle to calculate all other properties.

Side & Angle Calculator

Choose which side you have relative to the known angle

Side a
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Side b
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Hypotenuse
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Other Angle
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Trigonometry Functions
sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent

Remember: SOH CAH TOA

Right Triangle from Side & Area

Enter one side and the area to find all triangle properties.

Side & Area Calculator

Given one leg and area, find the other leg and hypotenuse: Area = ½ab

Other Leg
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Hypotenuse
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Angle 1
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Angle 2
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Area Formula
Area = ½ × base × height
Area = ½ × a × b

For right triangles, multiply the two legs and divide by 2. The two legs form the base and height.

Understanding Right Triangles

A right triangle is a triangle with exactly one 90-degree angle (right angle). The side opposite the right angle is called the hypotenuse and is always the longest side. The other two sides, called legs, form the right angle. This special property makes right triangles fundamental to mathematics, physics, engineering, and countless practical applications.

Right triangles are unique because their properties are completely determined by just two pieces of information—whether two sides, one side and an angle, or one side and the area. The Pythagorean theorem (a² + b² = c²) connects all three sides, and trigonometric functions (sine, cosine, tangent) relate sides to angles.

The 3-4-5 triangle is the most famous right triangle, representing the simplest Pythagorean triple. This triangle appears in construction, navigation, design, and problem-solving across virtually every discipline.

Key Property: In any right triangle, the sum of the two acute angles equals 90 degrees, and the Pythagorean theorem always holds: a² + b² = c².

Key Features & Capabilities

From Two Sides Calculate hypotenuse, angles, area, perimeter from both legs
From Side & Angle Find all sides and angles given one side and acute angle
From Side & Area Solve triangle from one leg and total area
Complete Triangle Get all 6 properties: 3 sides, 2 angles, area, perimeter
Trigonometry Uses sine, cosine, tangent for accurate calculations
Step-by-Step Detailed solution breakdown for every calculation

How to Use the Calculator

From Two Sides Method

  1. Measure Legs: Measure the two sides forming the 90-degree angle
  2. Enter Both: Input both measurements into the calculator
  3. Calculate: Press Calculate to solve
  4. Get Complete Triangle: Receive hypotenuse, both angles, area, and perimeter

From Side & Angle Method

  1. Know Side Length: Have at least one side measurement
  2. Know Angle: Know one acute angle (not the 90° angle)
  3. Specify Position: Indicate if side is opposite, adjacent, or the hypotenuse
  4. Calculate: Trigonometry finds all other values

From Side & Area Method

  1. Know One Leg: Have one of the leg measurements
  2. Know Area: Know the total triangle area
  3. Solve Other Leg: Use Area = ½ab to find second leg
  4. Complete Triangle: Calculate hypotenuse and angles

Important Notes

  • Right Angle: One angle is always 90°—don't include it in calculations
  • Acute Angles: The other two angles sum to 90°
  • Hypotenuse Longest: The hypotenuse is always the longest side
  • Positive Values: All measurements must be positive numbers
  • Angle Units: Use degrees (not radians) for angle input

Complete Formulas Reference

Pythagorean Theorem (Find Hypotenuse)
c² = a² + b²
c = √(a² + b²)
Find Missing Leg
a = √(c² - b²)
b = √(c² - a²)
Area of Right Triangle
Area = ½ × a × b
Area = (1/2) × base × height
Perimeter
P = a + b + c
P = a + b + √(a² + b²)
Trigonometric Ratios
sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent
Finding Angles
α = arctan(a/b) = arcsin(a/c) = arccos(b/c)
β = 90° - α
(The two acute angles sum to 90°)

Trigonometry in Right Triangles

Understanding SOHCAHTOA

The mnemonic SOHCAHTOA helps remember three fundamental trigonometric ratios:

  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent
Inverse Functions: To find angles from sides, use inverse trigonometric functions: arcsin (sin⁻¹), arccos (cos⁻¹), arctan (tan⁻¹). These give you the angle when you know the ratio.

Complementary Angles

In a right triangle, the two acute angles are complementary (sum to 90°). This means sin(α) = cos(90° - α) and vice versa. This relationship simplifies many trigonometric calculations.

Triangle Properties Summary

Property Formula/Relationship Explanation
Pythagorean Theorem a² + b² = c² Sum of squares of legs equals square of hypotenuse
Angle Sum α + β + 90° = 180° All angles in any triangle sum to 180°
Acute Angles α + β = 90° The two non-right angles are complementary
Area A = ½ab Half the product of the two legs (base × height ÷ 2)
Perimeter P = a + b + c Sum of all three sides
Hypotenuse c > a and c > b Hypotenuse is always the longest side

Worked Examples

Example 1: Classic 3-4-5 Triangle

Problem: Right triangle with legs 3 and 4. Find all properties.

Solution:
Hypotenuse: c = √(3² + 4²) = √25 = 5
Area: A = ½ × 3 × 4 = 6
Perimeter: P = 3 + 4 + 5 = 12
Angle α: tan⁻¹(3/4) ≈ 36.87°
Angle β: 90° - 36.87° ≈ 53.13°

Example 2: From Side and Angle

Problem: Hypotenuse of 10 and angle of 35°. Find all sides and other angle.

Solution:
Opposite leg: a = 10 × sin(35°) ≈ 5.74
Adjacent leg: b = 10 × cos(35°) ≈ 8.19
Other angle: 90° - 35° = 55°
Area: A = ½ × 5.74 × 8.19 ≈ 23.52
Perimeter: P ≈ 5.74 + 8.19 + 10 ≈ 23.93

Example 3: From Leg and Area

Problem: One leg is 6 and area is 15. Find the triangle.

Solution:
Area = ½ab, so 15 = ½ × 6 × b
Other leg: b = 5
Hypotenuse: c = √(6² + 5²) = √61 ≈ 7.81
Angle 1: tan⁻¹(5/6) ≈ 40.24°
Angle 2: 90° - 40.24° ≈ 49.76°

Example 4: Roof Pitch Problem

Problem: Roof spans 20 feet horizontally with 45° pitch. Find height and rafter length.

Solution:
For 45° angle: Height = (20/2) × tan(45°) = 10 feet
Rafter: c = 10 / sin(45°) ≈ 14.14 feet
This creates a 10-10-14.14 isosceles right triangle

Real-World Applications

Construction & Roofing

Builders use right triangle calculations for roof pitches, stair angles, ramp slopes, and diagonal bracing. The 3-4-5 rule ensures perfect right angles in foundations and framing.

Navigation & Surveying

Surveyors calculate distances using right triangle geometry. GPS positioning, sight lines, and terrain measurements all depend on right triangle calculations.

Engineering & Physics

Force vectors, velocity components, and many physics problems reduce to right triangle problems. Structural engineering uses trigonometry extensively for load calculations.

Art & Design

Perspective drawing, camera angles, and composition often involve right triangle principles. Aspect ratios and golden ratios are based on geometric triangles.

Computer Graphics & Gaming

3D graphics rendering, collision detection, lighting calculations, and physics simulations all rely on right triangle math and trigonometry.

Frequently Asked Questions

❓ What makes a triangle a right triangle?
A triangle is a right triangle if and only if it has exactly one 90-degree angle. The side opposite this angle is the hypotenuse. All right triangles satisfy the Pythagorean theorem.
❓ Can I solve a right triangle with just one measurement?
No, you need at least two independent measurements. Two sides, one side and one angle, or one side and the area will allow you to solve the triangle. One side alone is insufficient.
❓ Why is the hypotenuse always longest?
By the Pythagorean theorem, c² = a² + b² means c² is larger than both a² and b² individually, so c is larger than both a and b. Mathematically, the hypotenuse must be longest.
❓ What's the difference between sine, cosine, and tangent?
Sine uses opposite and hypotenuse, cosine uses adjacent and hypotenuse, tangent uses opposite and adjacent. Choose based on which sides you know: use sin/cos if you know hypotenuse, tan if you only know legs.
❓ Do the two acute angles always sum to 90°?
Yes, always. Since all triangle angles sum to 180° and one angle is 90°, the other two must sum to 90°. This is called complementary angles.
❓ How do I find an angle if I only have sides?
Use inverse trigonometric functions: arctan(opposite/adjacent), arcsin(opposite/hypotenuse), or arccos(adjacent/hypotenuse). Your calculator has sin⁻¹, cos⁻¹, tan⁻¹ buttons.

Solve Right Triangles Completely

Whether you're solving geometry homework, working on construction projects, designing structures, or understanding trigonometry, this comprehensive right triangle calculator solves all triangle properties from multiple input combinations with instant accuracy and detailed step-by-step solutions. Fast, reliable, completely free.

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