Right Triangle Formulas- Complete Mathematical Reference Guide
The Basics You Actually Need
A right triangle has one angle exactly 90 degrees. That's the whole deal. The side opposite that right angle is the hypotenuse—it's always the longest side. The other two sides are the legs.
Everything about right triangles flows from this setup. Master these concepts and you can solve any right triangle problem thrown at you.
The Pythagorean Theorem
This is the foundation. For any right triangle:
a² + b² = c²
Where a and b are the legs, and c is the hypotenuse.
That's it. Plug in two sides, solve for the third. Works every time.
Example
Leg a = 3, leg b = 4. What's the hypotenuse?
3² + 4² = c²
9 + 16 = c²
c² = 25
c = 5
Classic 3-4-5 triangle. You'll see it everywhere.
The Six Trig Ratios
Trigonometry sounds intimidating but it's just ratios. For a right triangle with an acute angle θ:
- sin(θ) = opposite ÷ hypotenuse
- cos(θ) = adjacent ÷ hypotenuse
- tan(θ) = opposite ÷ adjacent
The reciprocals exist too, in case you need them:
- csc(θ) = 1 ÷ sin(θ) = hypotenuse ÷ opposite
- sec(θ) = 1 ÷ cos(θ) = hypotenuse ÷ adjacent
- cot(θ) = 1 ÷ tan(θ) = adjacent ÷ opposite
Most problems only need sin, cos, and tan. Memorize those three first.
Inverse Trig Functions
When you know the ratio and need the angle, use inverse trig:
- θ = sin⁻¹(x) — arcsine
- θ = cos⁻¹(x) — arccosine
- θ = tan⁻¹(x) — arctangent
Your calculator has these. Look for "sin⁻¹" or "2nd" + trig function.
Special Right Triangles
Some triangles show up constantly. Memorize these ratios.
45-45-90 Triangle
Isosceles right triangle. The legs are equal.
Leg : Leg : Hypotenuse = 1 : 1 : √2
If each leg = 5, the hypotenuse = 5√2 ≈ 7.07
30-60-90 Triangle
Half an equilateral triangle. Common in geometry and physics.
Short Leg : Long Leg : Hypotenuse = 1 : √3 : 2
The short leg is opposite the 30° angle. The long leg is opposite the 60° angle.
Area Formulas
Two ways to find the area of a right triangle:
Method 1: (1/2) × base × height
Simplest when you know the two legs.
Method 2: (1/2) × hypotenuse × altitude to hypotenuse
Useful when the altitude is given or needs to be found.
For a right triangle, the altitude from the right angle to the hypotenuse creates two smaller similar triangles. That relationship gives you:
Altitude = (a × b) ÷ c
Where a and b are the legs, c is the hypotenuse.
Complete Formula Reference
| Formula | Use When |
|---|---|
| a² + b² = c² | Find any side using the other two |
| sin(θ) = opp ÷ hyp | Find angle or opposite side |
| cos(θ) = adj ÷ hyp | Find angle or adjacent side |
| tan(θ) = opp ÷ adj | Find angle or either leg |
| Area = ½ab | Find area using both legs |
| Altitude = (a×b)÷c | Find altitude to hypotenuse |
| c = a√2 (45-45-90) | Hypotenuse from equal legs |
| Long leg = short leg × √3 | 30-60-90 relationships |
How to Solve Any Right Triangle Problem
Follow this checklist:
- Identify what you know. Label the sides: which is the hypotenuse? Which are the legs?
- Identify what you need. Finding a side or an angle?
- Pick the right tool.
- Need a side? → Pythagorean theorem (if you have two sides) or trig ratios (if you have an angle)
- Need an angle? → Trig ratios and inverse trig functions
- Solve. Plug in numbers. Isolate the variable.
- Check your work. Does the answer make sense? Hypotenuse should be longest side. Angles should add to 180°.
Practice Problem
A right triangle has a leg of 8 and an angle of 35° adjacent to that leg. Find the hypotenuse.
You know: adjacent = 8, need hypotenuse
Use: cos(35°) = adjacent ÷ hypotenuse
cos(35°) = 8 ÷ hypotenuse
hypotenuse = 8 ÷ 0.8192
hypotenuse ≈ 9.77
Common Mistakes to Avoid
- Using the wrong side for trig ratios. "Opposite" and "adjacent" always relate to the angle you're working with—not some fixed side.
- Forgetting to check if your calculator is in degree mode. Radians give completely wrong answers for standard geometry problems.
- Confusing the altitude formula. Altitude = (leg × leg) ÷ hypotenuse works when the altitude drops from the right angle to the hypotenuse.
- Dropping the square in Pythagorean theorem. c² means c squared. Don't solve for c and forget to take the square root.
Quick Reference for Exams
When you're pressed for time:
- Draw the triangle. Always. Label hypotenuse, legs, and the target angle.
- SOHCAHTOA works for any right triangle: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.
- 3-4-5 and 5-12-13 are common Pythagorean triples. Know them on sight.
- If stuck, Pythagorean theorem + trig ratios solve everything.