Triangle basics
A triangle is a polygon that has three vertices. A vertex is a point where two or more curves, lines, or edges meet. In the case of a triangle, the vertices are joined by three line segments called edges. A triangle is usually referred to by its vertices, hence a triangle with vertices a, b, and c is typically denoted as Δabc.
Triangles are described based on the length of their sides and internal angles. A triangle in which all sides have equal lengths is called equilateral; a triangle with two equal sides is isosceles; a triangle with no equal sides is scalene. Tick marks on edges show equal lengths, while arcs show equal angles.
Triangles classified by internal angles fall into two categories: right or oblique. A right triangle is one in which one angle is 90° and the opposite side is the hypotenuse. An acute triangle has all angles less than 90°; an obtuse triangle has one angle greater than 90°.
Triangle facts, theorems, and laws
• A triangle cannot have more than one vertex with an internal angle ≥ 90°; otherwise it ceases to be a triangle.
• Interior angles add up to 180°. Exterior angles equal 180° minus the interior angle at the vertex.
• The sum of the lengths of any two sides of a triangle is always larger than the length of the third side.
• Pythagorean theorem (right triangles): a² + b² = c². Example: if a = 3 and c = 5, then b² = 25 − 9 = 16 → b = 4.
• Law of sines: a/sin(A) = b/sin(B) = c/sin(C). Example: if b = 2, B = 90°, C = 45°, then c = 2 × √2/2 × 1 = √2.
• Law of cosines (find angles from sides): A = arccos((b² + c² − a²)/(2bc)), B = arccos((a² + c² − b²)/(2ac)), C = arccos((a² + b² − c²)/(2ab)). Example: given a = 8, b = 6, c = 10 ⇒ B ≈ 36.87°.
Area of a triangle
• Using base and height: area = ½ · base · height. Example: base 6, height 5 → area = ½ × 5 × 6 = 15.
• Using two sides and the included angle: area = ½ ab sin(C) = ½ bc sin(A) = ½ ac sin(B). Example: a = 9, b = 7, C = 30° → area = ½ × 7 × 9 × sin(30°) = 15.75.
• Heron’s formula (requires all sides): area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2. Example: a = 3, b = 4, c = 5 ⇒ s = 6, area = √[6 × 3 × 2 × 1] = 6.
Median, inradius, and circumradius
• Median: a line segment from a vertex to the midpoint of the opposite side. Formulas: m_a = ½√(2b² + 2c² − a²), m_b = ½√(2a² + 2c² − b²), m_c = ½√(2a² + 2b² − c²). Example: a = 2, b = 3, c = 4 → m_a ≈ 3.391.
• Inradius (r): radius of the inscribed circle. r = area / s, where s = (a + b + c)/2. Example: if a = 3, b = 4, c = 5, then s = 6 and r = area/s = 6/6 = 1.
• Circumradius (R): radius of the circumscribed circle. R = a/(2 sin A) (and analogous formulas for other sides/angles).