Right Triangle Calculator: Properties, Formulas and Applications
What is a Right Triangle?
A right triangle (or right-angled triangle) is a triangle in which one angle is a right angle (90 degrees). The side opposite the right angle is called the hypotenuse, which is always the longest side of the triangle.
The other two sides are called the legs of the triangle. Right triangles have special properties that make them fundamental in geometry and trigonometry.
Properties and Formulas
In a right triangle with sides a, b, and hypotenuse c:
- Area: A = (1/2)ab
- Perimeter: P = a + b + c
- Angles: A + B + C = 180°, where C = 90° and A + B = 90°
- Altitude: The altitude to the hypotenuse creates two similar triangles
Additionally, the Pythagorean theorem relates the three sides: a² + b² = c²
Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Expressed as:
a² + b² = c²
This theorem is named after the ancient Greek mathematician Pythagoras, although the relationship was known to earlier civilizations. It's one of the most fundamental theorems in mathematics with numerous applications.
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Trigonometric Ratios
Right triangles are the foundation of trigonometry. For an angle θ in a right triangle:
- Sine (sin): sin(θ) = opposite/hypotenuse = a/c
- Cosine (cos): cos(θ) = adjacent/hypotenuse = b/c
- Tangent (tan): tan(θ) = opposite/adjacent = a/b
The reciprocal functions are:
- Cosecant (csc): csc(θ) = 1/sin(θ) = c/a
- Secant (sec): sec(θ) = 1/cos(θ) = c/b
- Cotangent (cot): cot(θ) = 1/tan(θ) = b/a
Real-World Applications
Right triangles have numerous practical applications:
- Construction: Ensuring corners are square (90°) and calculating roof slopes
- Navigation: Determining distances and bearings using triangulation
- Surveying: Measuring heights that can't be measured directly
- Architecture: Designing structures with right angles and calculating diagonal supports
- Engineering: Analyzing forces and designing mechanical components
- Astronomy: Calculating distances to celestial objects
- Computer Graphics: Rendering 3D objects and calculating lighting angles
Using Our Calculator
Our Right Triangle Calculator can compute all properties of a right triangle. Here's how to use it:
- Enter at least 2 values (at least one must be a side length)
- The calculator will determine all other properties of the right triangle
- Angle C is fixed at 90° for right triangles
- Side c is the hypotenuse (longest side, opposite the right angle)
- Click "Calculate Triangle" to see all results
- Trigonometric ratios for angle A are also displayed
The calculator uses trigonometric relationships and the Pythagorean theorem to find unknown values.
FAQs
Can a right triangle be isosceles?
Yes, a right triangle can be isosceles if the two legs (a and b) are equal in length. This is called an isosceles right triangle, where the two acute angles are both 45°.
Can a right triangle be equilateral?
No, a right triangle cannot be equilateral because all sides of an equilateral triangle are equal, which would make all angles equal to 60°, but a right triangle must have one 90° angle.
What is the hypotenuse?
The hypotenuse is the longest side of a right triangle, opposite the right angle. It's the side that doesn't form the right angle.
How can I tell if a triangle is a right triangle?
A triangle is a right triangle if one of its angles measures exactly 90°. Alternatively, if the sides satisfy the Pythagorean theorem (a² + b² = c²), then the triangle is a right triangle.