A 90 degree triangle, also known as a right triangle, is one of the most fundamental geometric shapes in mathematics. It consists of one angle that measures exactly 90 degrees, with the other two angles being acute. Right triangles are used in various fields such as architecture, construction, trigonometry, navigation, and physics.
90 Degree (Right) Triangle Calculator
🧮 What Is a 90 Degree Triangle?
A right triangle is a triangle in which one angle is exactly 90°. The side opposite the right angle is the hypotenuse, and the other two sides are known as the legs.
Properties:
- One 90° angle
- Two acute angles that sum to 90°
- Satisfies the Pythagorean Theorem: a² + b² = c²
- Allows use of trigonometric ratios (sine, cosine, tangent)
🔧 How to Use the 90 Degree Triangle Calculator
Step-by-Step Instructions:
- Select what you know:
You can choose two values, such as:- Two sides (a, b, or c)
- One angle and one side
- Height and base
- Enter the known values into the input fields
- Click “Calculate”
- The tool will compute:
- Missing side lengths (a, b, or hypotenuse c)
- Missing angles
- Triangle area
- Perimeter (optional)
📐 Triangle Formulas Used (Plain Text)
1. Pythagorean Theorem
Used when two sides are known:
a² + b² = c²
→ c = √(a² + b²)
→ a = √(c² − b²)
→ b = √(c² − a²)
2. Trigonometric Ratios
If you know one angle (θ) and one side:
- sin(θ) = opposite / hypotenuse
- cos(θ) = adjacent / hypotenuse
- tan(θ) = opposite / adjacent
Use these to find missing side lengths.
3. Area Formula
Area = (1/2) × base × height
4. Perimeter
Perimeter = a + b + c
🧪 Example Calculations
Example 1: Known legs a = 3, b = 4
Use Pythagorean Theorem:
c = √(3² + 4²) = √(9 + 16) = √25 = 5
Area = 0.5 × 3 × 4 = 6
Example 2: Known side a = 6, angle A = 30°
Use trigonometry:
- sin(30°) = a / c → 0.5 = 6 / c → c = 12
- Use Pythagorean Theorem to find b:
b = √(12² − 6²) = √(144 − 36) = √108 ≈ 10.39
Example 3: Hypotenuse = 10, angle B = 60°
- cos(60°) = adjacent / hypotenuse → 0.5 = b / 10 → b = 5
- sin(60°) = opposite / hypotenuse → √3/2 ≈ 0.866 = a / 10 → a ≈ 8.66
Area = 0.5 × 8.66 × 5 = 21.65
✅ Benefits of Using the Calculator
- ✅ Solves all triangle types with 90°: SSS, SAS, ASA combinations
- ✅ Accurate trigonometric calculations
- ✅ Time-saving: No manual equation solving required
- ✅ Supports decimals and whole numbers
- ✅ Useful for design, education, and construction
📊 Applications of 90 Degree Triangles
- Construction: stair design, slope measurements
- Trigonometry classes: solving practice problems
- Physics: resolving forces into components
- Navigation: finding distance across diagonals
- Surveying: land measurements and angles
📖 20 Frequently Asked Questions (FAQs)
1. What is a 90 degree triangle?
A triangle with one right angle (90°) and two acute angles.
2. What is the hypotenuse?
The side opposite the 90° angle; the longest side in a right triangle.
3. How do I find the hypotenuse?
Use the formula c = √(a² + b²).
4. Can this calculator find angles?
Yes, if at least one side and another angle or side is known.
5. What if I only know one side?
You’ll need at least one more value (another side or angle).
6. Can I input decimal values?
Yes, the calculator supports both decimal and whole number inputs.
7. What is the area of a right triangle?
Area = 0.5 × base × height
8. Can it calculate perimeter?
Yes, once all sides are found, it adds them up.
9. What if I input incorrect values?
The calculator may return an error or prompt for valid inputs.
10. Can it handle large numbers?
Yes, it’s suitable for academic and real-world use cases.
11. What if I know one angle and one side?
Use trigonometric ratios (sine, cosine, tangent) to find others.
12. Does it work on mobile devices?
Yes, it’s responsive and works on all screen sizes.
13. What’s the range for right triangle angles?
Besides the 90°, the other two angles must be less than 90° and together equal 90°.
14. Is this tool suitable for students?
Absolutely—it’s a great learning aid for geometry and trigonometry.
15. Does it explain steps?
Most versions show step-by-step calculations for educational purposes.
16. Can it find height?
Yes, when given the base and area or by using sine and cosine.
17. Is this calculator free?
Yes, it’s free to use online.
18. Can I use it for isosceles right triangles?
Yes, just enter the equal legs; the tool will do the rest.
19. Can I reverse-calculate sides from area?
Only if you know at least one side or angle.
20. What if all I know is the hypotenuse?
You need at least one leg or an angle to solve the triangle.
🧠 Final Thoughts
The 90 Degree Triangle Calculator is a versatile and user-friendly tool that helps students, professionals, and DIY enthusiasts solve right-angle triangle problems in seconds. With its support for trigonometry, Pythagorean calculations, and geometric formulas, it eliminates the guesswork and speeds up problem-solving.