Triangulo Rectangulo- Right Triangle Calculations and Formulas
What Is a Right Triangle?
A right triangle is a three-sided shape with one angle exactly 90 degrees. That corner with the square symbol? That's the right angle. The side opposite the right angle is the longest one — called the hypotenuse. The other two sides are the legs.
That's it. That's the whole shape. Simple geometry that shows up everywhere from construction to video games.
The Parts You Need to Know
- Hypotenuse — the side across from the right angle. Always the longest side.
- Legs — the two sides that form the right angle. One is adjacent to the angle you're working with, one is opposite.
- Right angle — 90 degrees. Marked with a small square in diagrams.
The Pythagorean Theorem
This is the main formula for right triangles:
a² + b² = c²
Where a and b are the legs, and c is the hypotenuse.
Use it when you know two sides and need the third. That's it. That's the entire practical application.
Example
Leg a = 3, leg b = 4. What's the hypotenuse?
3² + 4² = c²
9 + 16 = c²
25 = c²
c = 5
Classic 3-4-5 triangle. Works every time.
Trigonometric Ratios
When you need to find angles or specific sides, trigonometry kicks in. Three ratios cover everything:
Sine (sin)
sin(θ) = opposite / hypotenuse
Use when you know the angle and hypotenuse, and need the opposite side. Or when you know the opposite and hypotenuse, and need the angle.
Cosine (cos)
cos(θ) = adjacent / hypotenuse
Use when you know the angle and hypotenuse, and need the adjacent side.
Tangent (tan)
tan(θ) = opposite / adjacent
Use when you don't have the hypotenuse. Just two legs and an angle.
SOH CAH TOA Memory Trick
Nobody memorizes the full words. Use this instead:
- SOH — Sine = Opposite / Hypotenuse
- CAH — Cosine = Adjacent / Hypotenuse
- TOA — Tangent = Opposite / Adjacent
Write it once. Say it three times. You'll remember it for years.
Area of a Right Triangle
No special formula needed. Use the standard area formula — the right angle makes it easy:
Area = (leg × leg) / 2
Multiply the two legs, divide by 2. That's the whole calculation.
Example
Legs are 6 and 8.
Area = (6 × 8) / 2 = 48 / 2 = 24
Inverse Trigonometric Functions
Sometimes you know the sides and need the angle. That's what arcsin, arccos, and arctan are for.
- θ = sin⁻¹(opp/hyp) — gives angle from opposite and hypotenuse
- θ = cos⁻¹(adj/hyp) — gives angle from adjacent and hypotenuse
- θ = tan⁻¹(opp/adj) — gives angle from both legs
Your calculator has these. Look for the "2nd" or "shift" key, then the sin/cos/tan button.
Common Right Triangle Ratios
These show up constantly. Memorize them and skip the calculator for standard angles:
| Angle | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Formula Reference Table
| What You Know | What You Need | Formula |
|---|---|---|
| Both legs | Hypotenuse | c = √(a² + b²) |
| Hypotenuse + one leg | Other leg | a = √(c² - b²) |
| Angle + hypotenuse | Opposite side | opp = hyp × sin(θ) |
| Angle + hypotenuse | Adjacent side | adj = hyp × cos(θ) |
| Both legs | Angle | θ = tan⁻¹(opp/adj) |
| Both legs | Area | (a × b) / 2 |
How to Solve a Right Triangle
Step 1: Identify What You Have
Count your known values. Two sides? Two angles? One side and one angle? The path forward depends entirely on this.
Step 2: Choose Your Formula
Pythagorean theorem for missing sides. Trigonometry for missing angles. Area formula if that's what the problem asks.
Step 3: Plug In and Solve
Substitute your numbers. Calculate. Check your work — if you used Pythagorean, verify that a² + b² equals c².
Step 4: Find Remaining Values
With most problems, finding one missing piece lets you find the rest. The triangle angles always sum to 180°, and you already have one 90° angle.
Where Right Triangles Actually Show Up
- Construction — roof pitches, stair angles, leveling foundations
- Surveying — measuring distances without crossing obstacles
- Navigation — compass bearings and distance calculations
- Computer graphics — rendering, collision detection, movement vectors
- Architecture — structural load calculations, diagonal measurements
Quick Calculator Reference
If you're doing this by hand, here's what to type on any basic calculator:
- Find hypotenuse: √(a² + b²)
- Find angle from sides: tan⁻¹(opp/adj) — gives angle in degrees
- Find side from angle: sin(θ) × hyp — use 30, 45, or 60 for quick answers
The Bottom Line
Right triangles are simple. One right angle. Two legs. One hypotenuse. Three formulas cover almost everything: Pythagorean theorem for sides, sin/cos/tan for angles, and (a×b)/2 for area.
Everything else in geometry builds on this. Master these basics first.