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.
Key Formula
Area = (1/2) × base × height
LaTeX: A = \frac{1}{2} b h
| Symbol | Meaning | Unit |
|---|---|---|
| A | area of the triangle | square units |
| b | length of the base | units |
| h | perpendicular height from base to opposite vertex | units |
Worked Example
Problem
Find the area of a triangle with base 10 cm and height 6 cm. Also verify that its angles 50°, 70°, and 60° are valid.
Solution
Step 1: Area = (1/2) × base × height = (1/2) × 10 × 6 = 30 cm². Step 2: Check angles: 50° + 70° + 60° = 180°. ✓ Valid triangle.
Answer
Area = 30 cm²; angle sum = 180° (valid)
Types of Triangles by Side Length and Angle
| Classification | Type | Property | Example Angles |
|---|---|---|---|
| By sides | Equilateral | All 3 sides equal | 60°, 60°, 60° |
| By sides | Isosceles | 2 sides equal | 50°, 65°, 65° |
| By sides | Scalene | All sides different | 40°, 60°, 80° |
| By angles | Acute | All angles < 90° | 60°, 70°, 50° |
| By angles | Right | One angle = 90° | 90°, 45°, 45° |
| By angles | Obtuse | One angle > 90° | 120°, 30°, 30° |
Interactive Tools
Wikimedia Commons, CC BY-SA
Related Terms
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 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.
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.
From Latin "triangulum" meaning three-angled figure, combining "tri-" (three) from Greek "treis" and "angulus" (angle). The word has been used in English since the late 14th century. Greek used "trigonon" (τρίγωνον), which survives in "trigonometry".