Right Triangle
A right triangle has one 90° angle. The side opposite the right angle is the hypotenuse (always longest). Solve from two sides using the Pythagorean theorem, or from one side and one acute angle using SOHCAHTOA.
Why This Mathematical Concept Matters
Why: Right triangles are foundational for construction, surveying, navigation, and physics. Pythagorean triples like 3-4-5 have been used since ancient Egypt.
How: Two sides: use a²+b²=c² to find the third. One side + angle: use sin, cos, tan. Area = ½×base×height. Special triangles (30-60-90, 45-45-90) have fixed ratios.
- ●The ancient Egyptians used 3-4-5 triangles to create perfect right angles when building pyramids.
- ●GPS triangulation relies on right triangle geometry to pinpoint your location.
- ●Roof pitch is expressed as rise-over-run—a 6/12 pitch forms a right triangle.
Right Triangles — Pythagorean Theorem & Trig Ratios
Solve any right triangle from two sides or one side + angle. Special triples (3-4-5, 5-12-13), 30-60-90, 45-45-90.
📐 Common Right Triangle Examples — Click to Load
Input Mode
Which Two Sides?
Enter Values
Right Triangle Visualization
Triangle Properties Radar
Side Length Comparison
Angle Proportions
Step-by-Step Breakdown
Step 1: A right triangle has one angle that is exactly 90 degrees (the right angle).
Step 2: Given the base and height, we can calculate the hypotenuse using the Pythagorean theorem:
Step 3: Substitute the values:
Step 4: Calculate the hypotenuse:
Step 5: Calculate the area of the right triangle:
Step 6: Calculate the perimeter of the right triangle:
⚠️For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
Ancient Egyptians used 3-4-5 right triangles to create perfect 90° angles.
— Khan Academy
The Pythagorean theorem predates Pythagoras—Babylonian tablets from 1800 BCE show it.
— Wolfram MathWorld
Key Takeaways
- The Pythagorean theorem (a² + b² = c²) relates the legs and hypotenuse — the hypotenuse is always opposite the 90° angle
- With any two sides known, you can find the third; with one side and one acute angle, trig ratios (sin, cos, tan) solve the rest
- Special right triangles: 30-60-90 has ratio 1:√3:2; 45-45-90 has ratio 1:1:√2
- Pythagorean triples like (3,4,5), (5,12,13), (8,15,17) are whole-number solutions — used in construction and surveying
- Right triangles are foundational for construction, navigation, and surveying
Did You Know?
How Right Triangle Calculations Work
Right triangles have one 90° angle. The side opposite the right angle is the hypotenuse (always the longest). The other two sides are the legs (base and height).
Two Sides Mode — Pythagorean Theorem
Given any two sides, use a² + b² = c² to find the third. If base and height: c = √(a² + b²). If base and hypotenuse: b = √(c² − a²). If height and hypotenuse: a = √(c² − b²). See our Pythagorean Theorem Calculator for verification.
Side and Angle Mode — Trig Ratios
sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent. With one side and one acute angle, these ratios yield the other sides. Use our 30-60-90 Calculator or 45-45-90 Calculator for special triangles.
Area and Perimeter
Area = ½ × base × height (the legs form a natural base-height pair). Perimeter = base + height + hypotenuse. For area from sides only, see our Triangle Area Calculator.
Expert Tips for Right Triangles
Hypotenuse Is Always Longest
If your "hypotenuse" is shorter than a leg, you've swapped values. The hypotenuse is opposite the 90° angle and must be the longest side.
Memorize Pythagorean Triples
(3,4,5), (5,12,13), (8,15,17), (7,24,25) — these whole-number triples appear in exams and real-world problems. Scale them: 6-8-10 is 2× the 3-4-5.
Special Triangles Save Time
30-60-90: sides 1:√3:2. 45-45-90: sides 1:1:√2. Recognizing these lets you skip trig in many problems.
Angle Must Be Acute (0° < θ < 90°)
In side+angle mode, the angle is one of the two non-right angles. It must be strictly between 0 and 90 degrees.
Why Use This Calculator vs. Other Tools?
| Feature | This Calculator | Wolfram Alpha | Manual Calculation |
|---|---|---|---|
| Two sides OR side + angle input | ✅ | ✅ | ⚠️ Tedious |
| Step-by-step solutions | ✅ | ⚠️ Paid | ❌ |
| Interactive visualization | ✅ | ✅ | ❌ |
| Pythagorean triples examples | ✅ | ✅ | ❌ |
| Charts (radar, bar, doughnut) | ✅ | ❌ | ❌ |
| Copy & share results | ✅ | ❌ | ❌ |
| Explain with AI | ✅ | ❌ | ❌ |
| Free (no signup) | ✅ | ⚠️ Limited | ✅ |
Frequently Asked Questions
How do I know if three sides form a right triangle?
Check if a² + b² = c² where c is the longest side. For example, 3² + 4² = 9 + 16 = 25 = 5², so (3,4,5) is a right triangle.
What are Pythagorean triples?
Sets of three positive integers satisfying a² + b² = c². Common ones: (3,4,5), (5,12,13), (8,15,17), (7,24,25). They scale: (6,8,10) is 2× (3,4,5).
Can a right triangle have two equal sides?
Yes — an isosceles right triangle (45-45-90). Both legs are equal, and the hypotenuse = leg × √2.
Why is the hypotenuse always the longest side?
It's opposite the 90° angle — the largest angle in any triangle. Larger angles are opposite longer sides.
When do I use trig vs. Pythagorean theorem?
Use Pythagorean theorem when you have two sides. Use sin/cos/tan when you have one side and one acute angle.
What is the 30-60-90 triangle ratio?
Sides are in ratio 1 : √3 : 2 (short leg : long leg : hypotenuse). If the short leg is 5, the long leg is 5√3 and hypotenuse is 10.
What is the 45-45-90 triangle ratio?
Sides are 1 : 1 : √2. Both legs are equal; the hypotenuse is leg × √2.
How accurate is this calculator?
Uses IEEE 754 double-precision (~15 significant digits). Displayed to 4 decimal places — more than sufficient for construction, surveying, and education.
Right Triangle by the Numbers
Official & Trusted Sources
Disclaimer: This calculator provides mathematically precise results based on the Pythagorean theorem and trigonometric ratios. Results are limited by floating-point precision (~15 significant digits). For critical engineering, surveying, or construction applications, verify with professional tools. Not a substitute for licensed surveying or structural engineering.