Right Triangle Calculator
Solve a right triangle given any two values. Calculate all sides, angles, area, and perimeter of a right triangle. Solve right triangles using the Pythagorean theorem. Find unknown sides and angles when two measurements are known.
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Right Triangle Calculator: Solve All Sides and Angles
The Right Triangle Calculator is an ultimate geometry solver that fully defines a right-angled triangle from any two known measurements. Whether you know two sides, or one side and one acute angle, this tool instantly calculates the missing side, both acute angles, the area, and the perimeter.
Right triangles are the cornerstone of trigonometry and appear everywhere in the physical world. From ensuring building corners are perfectly square, to determining the height of a tree using a shadow, to calculating vector components in physics, the right triangle is an indispensable mathematical model.
By automating the Pythagorean theorem and inverse trigonometric functions (SOH CAH TOA), this calculator eliminates the need to look up formulas, convert radians to degrees, or manually manipulate complex equations.
💡 Geometry Rule: To completely solve a right triangle, you only need two pieces of information (aside from knowing one angle is 90°). However, at least one of those pieces must be a side length. Knowing only two angles gives you the shape, but not the size (scale).
When to Use This Right Triangle Calculator
🔨 Construction & Carpentry
Calculate rafter lengths (hypotenuse) from roof span and pitch. Ensure foundations are perfectly square using the 3-4-5 rule.
⚙️ Machining & Fabrication
Calculate tool paths, chamfer depths, and taper angles. Convert between cartesian (x,y) and polar (angle, length) coordinates.
🎓 Education
Check homework for trigonometry, geometry, and physics. Master the relationship between sine, cosine, tangent, and side lengths.
🧭 Navigation & Surveying
Determine straight-line distance across impassable terrain using triangulation. Calculate heading and distance from North/East offsets.
📐 Architecture
Determine staircase stringer lengths based on tread and riser dimensions. Calculate ADA ramp lengths from required slopes.
🍎 Physics (Vectors)
Break down force, velocity, or acceleration vectors into vertical (y) and horizontal (x) components, or combine them into a resultant vector.
Key Right Triangle Formulas
The Pythagorean Theorem
Relates the three sides of the right triangle:
a² + b² = c²
Where c is the hypotenuse (the longest side opposite the 90° angle), and a and b are the legs.
Trigonometric Ratios (SOH CAH TOA)
Relates the angles to the side lengths. For any acute angle θ:
- Sine (sin): sin(θ) = Opposite / Hypotenuse
- Cosine (cos): cos(θ) = Adjacent / Hypotenuse
- Tangent (tan): tan(θ) = Opposite / Adjacent
Area and Perimeter
Area = ½ × a × b
Perimeter = a + b + c
Step-by-Step Guide: Solving from One Leg and One Angle
Given: Leg (adjacent) a = 10 m, Angle α = 30°. We need to find Leg b (opposite), Hypotenuse c, and Angle β.
Find the other angle: The acute angles sum to 90°. β = 90° − 30° = 60°.
Find Leg b (opposite): Use tangent (TOA). tan(30°) = b / 10.
b = 10 × tan(30°) ≈ 10 × 0.5774 = 5.774 m.
Find Hypotenuse c: Use cosine (CAH). cos(30°) = 10 / c.
c = 10 / cos(30°) ≈ 10 / 0.866 = 11.547 m.
Check with Pythagoras: 10² + 5.774² = 100 + 33.33 = 133.33. And 11.547² ≈ 133.33. The math checks out!
Calculate Area: Area = ½ × 10 × 5.774 = 28.87 m².
5 Worked Examples
Example 1: The 3-4-5 Triangle
Known: Leg a = 3, Leg b = 4
c = √(3² + 4²) = √(9+16) = √25 = 5. Area = ½ × 3 × 4 = 6. Angle A = arctan(3/4) ≈ 36.87°. Angle B = arctan(4/3) ≈ 53.13°.
Example 2: Isosceles Right Triangle
Known: Leg a = 10, Angle A = 45°
Since A=45°, B=45°. Since angles are equal, legs are equal: b = 10. Hypotenuse c = √(10²+10²) = 10√2 ≈ 14.14. Area = ½ × 10 × 10 = 50.
Example 3: Finding a Ladder Angle
Known: A 12 ft ladder (hypotenuse c = 12) reaches a window 10 ft high (leg a = 10).
Base distance from wall: b = √(12² - 10²) = √(144 - 100) = √44 ≈ 6.63 ft. Angle of ladder with ground: sin(A) = 10/12. A = arcsin(0.833) ≈ 56.4°.
Example 4: Surveying a River Width
Known: Base leg along bank b = 50 m. Angle to target on opposite bank A = 40°.
River width (opposite leg a) = 50 × tan(40°) ≈ 50 × 0.839 = 41.95 m. Distance across water (hypotenuse) = 50 / cos(40°) ≈ 65.27 m.
Example 5: Wheelchair Ramp
Known: Required height (rise a) = 2 ft. Slope must be 1:12 (approx 4.76°).
Run (leg b): The ratio is 1:12, so b = 12 × a = 24 ft. Ramp surface length (hypotenuse c) = √(2² + 24²) = √(4 + 576) = √580 ≈ 24.08 ft.
Common Mistakes to Avoid
❌ Adding Instead of Subtracting in Pythagoras
Consequence: When finding a leg, many mistakenly calculate a² + c² instead of c² - a². This results in a leg longer than the hypotenuse, which is physically impossible.
✓ Fix: Remember: Hypotenuse = √(a² + b²). Leg = √(c² - a²). The hypotenuse MUST be the longest side.
❌ Radians vs. Degrees
Consequence: Performing trigonometric calculations with a calculator set to radians when you entered degrees yields wildly incorrect answers.
✓ Fix: This calculator automatically handles degrees. If doing manual math, always check your calculator's mode (DEG/RAD).
❌ Confusing Adjacent and Opposite Legs
Consequence: Using sine instead of cosine because the legs were swapped. The "opposite" leg is across from the angle you are looking at. The "adjacent" leg touches the angle.
✓ Fix: Draw the triangle. Label the hypotenuse first. Then label the opposite and adjacent legs relative to your specific angle of interest.
Tips and Best Practices
- Memorize common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Any multiple of these is also a right triangle (e.g., 6-8-10).
- Check your work: The largest angle (90°) must be opposite the longest side (hypotenuse). The smallest angle must be opposite the shortest leg.
- To square a corner in real life: Measure 3 units along one wall, 4 units along the adjacent wall. Adjust the walls until the diagonal distance between those two marks is exactly 5 units.
- Use the Area formula two ways: Area = ½(a)(b). It also equals ½(c)(h) where h is the altitude to the hypotenuse. You can use this to find the altitude: h = (a×b)/c.
Frequently Asked Questions
▶What defines a right triangle?
▶Can a right triangle have an obtuse angle?
▶How many values do I need to solve the triangle?
▶What is SOH CAH TOA?
▶Why is the hypotenuse always the longest side?
▶Can the two legs be equal?
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Conclusion
The Right Triangle Calculator is an essential tool for anyone working with geometry, construction, or physics. By instantly applying the Pythagorean theorem and SOH CAH TOA trigonometric ratios, it takes any two known measurements and reveals the entire geometry of the triangle.
Whether you are squaring up a deck foundation, solving a vector physics problem, or simply checking your math homework, this tool guarantees precision. Always ensure you are entering side lengths in the same units and remember that the hypotenuse must always be your longest side.
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