Right Triangle Calculator
Solve right triangles using the Pythagorean theorem and trigonometry
How to Use This Right Triangle Calculator
- Choose your input method: Two Sides, Side + Angle, or Hypotenuse + Angle
- Enter the known values (all angles are in degrees)
- For two sides: select which sides (a, b, or c) you're entering
- For side + angle: specify if the side is adjacent or opposite to the angle
- Click 'Solve Triangle' to find all missing sides, angles, and properties
Example: A ladder leans against a wall at 75 degrees, with base 3 feet from the wall. Enter: Side a (adjacent) = 3, Angle = 75 degrees. Results: ladder length (c) = 11.59 ft, wall height reached (b) = 11.20 ft.
Tip: Remember SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Why Use a Right Triangle Calculator?
Right triangles are fundamental to navigation, construction, physics, and engineering calculations.
- Calculate roof pitch and rafter lengths for construction projects
- Determine safe ladder angles and reach heights
- Find distances using indirect measurement (surveying technique)
- Solve physics problems involving vectors and components
- Calculate grade percentages for roads, ramps, and slopes
- Plan sight lines and angles for photography and architecture
Understanding Your Results
The calculator provides all three sides, both acute angles, trigonometric ratios, area, and perimeter.
| Result | Meaning | Action |
|---|---|---|
| 45-45-90 triangle | Isosceles right triangle | Legs are equal; hypotenuse = leg x sqrt(2) |
| 30-60-90 triangle | Special right triangle | Sides in ratio 1 : sqrt(3) : 2 |
| Integer sides (3-4-5, etc.) | Pythagorean triple | Exact measurements - no rounding errors |
Meaning: Isosceles right triangle
Action: Legs are equal; hypotenuse = leg x sqrt(2)
Meaning: Special right triangle
Action: Sides in ratio 1 : sqrt(3) : 2
Meaning: Pythagorean triple
Action: Exact measurements - no rounding errors
Note: The two acute angles always sum to 90 degrees (since the right angle is 90 degrees and triangle angles total 180).
About Right Triangle Calculator
Formula
sin(theta) = opposite/hypotenuse | cos(theta) = adjacent/hypotenuse | tan(theta) = opposite/adjacent Given any side and acute angle, trigonometry finds the other sides. Given two sides, inverse trig (arcsin, arccos, arctan) finds angles.
Current Standards: Angles can be measured in degrees (360 per circle) or radians (2*pi per circle). This calculator uses degrees.
Frequently Asked Questions
What's the difference between this and the Pythagorean theorem calculator?
The Pythagorean calculator only uses a^2 + b^2 = c^2 to find a missing side when you know two sides. This calculator adds trigonometry, so you can solve triangles knowing one side and one angle. It's more versatile but requires understanding which angle references which side.
How do I know which side is adjacent vs opposite?
It depends on which angle you're referencing. The side touching the angle (not the hypotenuse) is adjacent. The side across from the angle is opposite. The hypotenuse is always opposite the right angle and touches both acute angles.
Why do my trig calculations seem wrong on some calculators?
Check your angle mode! Scientific calculators and programming languages often default to radians. If you enter 45 expecting degrees but the calculator uses radians, results will be wrong. This calculator uses degrees.
What are common angle relationships I should memorize?
sin(30) = 0.5, cos(30) = sqrt(3)/2 = 0.866, tan(30) = 1/sqrt(3) = 0.577. sin(45) = cos(45) = sqrt(2)/2 = 0.707, tan(45) = 1. sin(60) = cos(30), cos(60) = sin(30), tan(60) = sqrt(3) = 1.732.
How do I use this for grade/slope calculations?
Grade is rise over run (vertical change over horizontal change), which equals tan(angle). A 10% grade means 10 ft rise per 100 ft horizontal = tan(theta) = 0.1, so theta = arctan(0.1) = 5.7 degrees. Enter the angle and one distance to find the other.