Right Triangle Calculator - Calculate Sides, Angles, Area & Perimeter
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The right angle (90°) is at vertex C. Angle A + Angle B = 90°.
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Understanding Right Triangles
What is a Right Triangle?
A right triangle is a triangle with one angle measuring exactly 90 degrees (a right angle). This special property makes right triangles fundamental in geometry, trigonometry, and real-world applications. The side opposite the right angle is called the hypotenuse (c), which is always the longest side. The other two sides are called legs (a and b).
The right angle is typically denoted by a small square in the corner at vertex C. The two acute angles (A and B) always sum to 90 degrees, making them complementary angles.
The Pythagorean Theorem
The Pythagorean theorem is the cornerstone of right triangle calculations. It states that the square of the hypotenuse equals the sum of the squares of the other two sides:
a² + b² = c²
This theorem allows you to find any side of a right triangle if you know the other two sides. For example:
- If a = 3 and b = 4, then c = √(3² + 4²) = √(9 + 16) = √25 = 5
- If a = 5 and c = 13, then b = √(13² - 5²) = √(169 - 25) = √144 = 12
Trigonometric Functions
Right triangles are the basis for trigonometry. The three primary trigonometric functions relate the angles to the side ratios:
Sine (sin)
sin(A) = opposite / hypotenuse
sin(A) = a / c
Cosine (cos)
cos(A) = adjacent / hypotenuse
cos(A) = b / c
Tangent (tan)
tan(A) = opposite / adjacent
tan(A) = a / b
These functions allow you to find angles when you know the sides, or find sides when you know an angle. Use inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) to find angles from side ratios.
Special Right Triangles
45-45-90 Triangle
Also called an isosceles right triangle. Both legs are equal in length.
- Angles: 45°, 45°, 90°
- Side ratio: 1 : 1 : √2
- If leg = x, then hypotenuse = x√2
30-60-90 Triangle
Half of an equilateral triangle. Has convenient side ratios.
- Angles: 30°, 60°, 90°
- Side ratio: 1 : √3 : 2
- If shortest side = x, then hypotenuse = 2x
Real-World Applications
- Construction: Ensuring walls are perpendicular and roofs have correct pitch
- Navigation: Calculating distances and bearings between locations
- Physics: Resolving forces and velocities into components
- Surveying: Measuring heights and distances indirectly
- Architecture: Designing stable structures and determining dimensions
Pro Tips
- • Always verify your answer using the Pythagorean theorem
- • Remember that the hypotenuse is always the longest side
- • The two acute angles always sum to 90°
- • Use inverse trig functions to find angles from side ratios
- • Special triangles (30-60-90, 45-45-90) have memorizable ratios