Right Triangle
A right triangle is a triangle containing exactly one right angle (90°). The side opposite the right angle is called the hypotenuse and is always the longest side, while the other two sides are called legs or catheti. Right triangles are the foundation of trigonometry and appear throughout architecture, engineering, and physics in the analysis of forces, distances, and angles.
Key Formula
sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent
LaTeX: \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}
| Symbol | Meaning | Unit |
|---|---|---|
| θ | one of the non-right angles in the triangle | degrees or radians |
| opposite | side opposite to angle θ | units |
| adjacent | side adjacent to angle θ (not the hypotenuse) | units |
| hypotenuse | side opposite the right angle | units |
Worked Example
Problem
A right triangle has legs of length 3 cm and 4 cm. Find the hypotenuse and the sine, cosine, and tangent of the angle opposite the 3 cm leg.
Solution
Step 1: Hypotenuse c = √(3² + 4²) = √(9 + 16) = √25 = 5 cm. Step 2: Let θ be the angle opposite the 3 cm leg. Step 3: sin θ = opposite/hypotenuse = 3/5 = 0.6. Step 4: cos θ = adjacent/hypotenuse = 4/5 = 0.8. Step 5: tan θ = opposite/adjacent = 3/4 = 0.75.
Answer
Hypotenuse = 5 cm; sin θ = 0.6, cos θ = 0.8, tan θ = 0.75
Special Right Triangles and Their Ratios
| Type | Angles | Side Ratios | Example Sides |
|---|---|---|---|
| 45-45-90 | 45°, 45°, 90° | 1 : 1 : √2 | 5 cm, 5 cm, 5√2 ≈ 7.07 cm |
| 30-60-90 | 30°, 60°, 90° | 1 : √3 : 2 | 5 cm, 5√3 ≈ 8.66 cm, 10 cm |
| 3-4-5 | approx 37°, 53°, 90° | 3 : 4 : 5 | 3 cm, 4 cm, 5 cm |
| 5-12-13 | approx 23°, 67°, 90° | 5 : 12 : 13 | 5 cm, 12 cm, 13 cm |
Interactive Tools
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Related Terms
Pythagorean Theorem
The Pythagorean Theorem states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides (the legs). It is one of the most famous and widely applied theorems in mathematics, used in distance calculations, navigation, construction, and virtually every branch of science and engineering.
Triangle
A triangle is a polygon with exactly three sides, three angles, and three vertices. The sum of the interior angles of any triangle always equals 180°, making it the simplest closed polygon. Triangles are the most rigid of all polygons and are widely used in engineering structures, architecture, and navigation due to their inherent stability.
Angle
An angle is the measure of rotation between two rays (sides) that share a common endpoint called the vertex. Angles are measured in degrees (°) or radians (rad) and describe the amount of turn between two directions. They are fundamental to geometry, trigonometry, physics, and engineering, appearing in everything from architectural blueprints to robotic arm movements.
"Right" comes from Old English "riht" meaning straight, direct, or upright, from Proto-Germanic "rehtaz". In geometry, a "right angle" means a perpendicular or exactly upright angle. "Triangle" is from Latin "triangulum" (three-angled). The concept of the right angle as 90° was systematized by ancient Greek geometers.