Triangle

A triangle is a three-sided polygon. It is the simplest closed shape in Euclidean geometry.

h a b c

Classification

Triangles can be classified by side length: equilateral (all sides equal), isosceles (two sides equal), scalene (no sides equal). By angle: acute (all angles < 90°), right (one angle = 90°), obtuse (one angle > 90°).

Basic Formulas

S = ½ · a · h Area using base a and height h
P = a + b + c Perimeter as sum of all sides
S = √[p(p-a)(p-b)(p-c)] Heron's formula (semi-perimeter method)

Right Triangle

For a right triangle with legs a, b and hypotenuse c:

a² + b² = c² Pythagorean theorem
S = ½ · a · b Area (legs as base and height)

Example

Given: base a = 8 cm, height h = 5 cm, sides b = 7 cm, c = 6 cm

Area: S = ½ · 8 · 5 = 20 cm²

Perimeter: P = 8 + 7 + 6 = 21 cm

Semi-perimeter: p = 21 ÷ 2 = 10.5 cm

Heron area: S = √[10.5(10.5-8)(10.5-7)(10.5-6)] = √[10.5·2.5·3.5·4.5] = √413.4375 ≈ 20.333 cm²

Triangle Inequality

For any triangle with sides a, b, c: each side must be less than the sum of the other two sides. If this condition is not met, a triangle cannot exist.

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