Understanding Triangles: Types, Properties, and Formulas

A triangle is a polygon with three vertices, where a vertex is the point where two or more edges meet. Triangles are defined by their vertices and the line segments (edges) connecting them. For example, a triangle with vertices A, B, C is denoted as ΔABC. Triangles can be classified based on the lengths of their sides and the measures of their internal angles.

Types of Triangles by Sides

• Equilateral triangle: All three sides are equal in length.
• Isosceles triangle: Two sides are equal in length.
• Scalene triangle: All three sides have different lengths.

Types of Triangles by Angles

• Right triangle: One angle is exactly 90°. The side opposite the right angle is called the hypotenuse.
• Oblique triangle: No right angle. Can be:
  • Acute triangle: All angles less than 90°.
  • Obtuse triangle: One angle greater than 90°.

Triangle Theorems and Laws

• The sum of the interior angles of a triangle is always 180°.
• The sum of the lengths of any two sides is always greater than the third side.

Pythagorean Theorem

Specific to right triangles, the square of the hypotenuse equals the sum of the squares of the other two sides:

a² + b² = c²

Law of Sines

The ratio of a side to the sine of its opposite angle is constant:

a / sin(A) = b / sin(B) = c / sin(C)

Angles from Side Lengths

Given sides a, b, c:

A = arccos((b² + c² - a²) / 2bc)
B = arccos((a² + c² - b²) / 2ac)
C = arccos((a² + b² - c²) / 2ab)

Area of a Triangle

• Using base and height: Area = 1/2 × base × height
• Using two sides and included angle: Area = 1/2 × a × b × sin(C)
• Using Heron’s formula: Area = √[s(s - a)(s - b)(s - c)] where s = (a + b + c)/2

Medians, Inradius, and Circumradius

Median

The median is the line segment from a vertex to the midpoint of the opposite side. All three medians intersect at the centroid of the triangle.

Inradius

The inradius is the radius of the largest circle that fits inside the triangle, calculated as:

Inradius = Area / Semiperimeter
Semiperimeter, s = (a + b + c) / 2

Circumradius

The circumradius is the radius of the circle passing through all vertices. It is calculated as:

Circumradius = a / (2 × sin(A))

Any side and its opposite angle can be used in this formula.