Triangle Calculator

About Triangles

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 three 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.

Types of Triangles

Triangles can be classified based on the length of their sides:

• Equilateral Triangle: All three sides have equal lengths
• Isosceles Triangle: Two sides have equal lengths
• Scalene Triangle: None of the sides have equal lengths

Triangles can also be classified based on their internal angles:

• Right Triangle: One of the angles is 90°
• Obtuse Triangle: One of the angles is greater than 90°
• Acute Triangle: All of the angles are less than 90°

Triangle Properties

Here are some important properties of triangles:

• The interior angles of a triangle always add up to 180°
• The sum of the lengths of any two sides of a triangle is always larger than the length of the third side
• It is not possible for a triangle to have more than one vertex with an internal angle greater than or equal to 90°

Triangle Formulas

Area of a Triangle

There are multiple formulas for calculating the area of a triangle:

1. Using base and height: Area = ½ × base × height
2. Using two sides and the included angle: Area = ½ × a × b × sin(C)
3. Using three sides (Heron's formula): Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2

Pythagorean Theorem

For right triangles: a² + b² = c², where c is the hypotenuse.

Law of Sines

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

Law of Cosines

a² = b² + c² - 2bc × cos(A)

Median, Inradius, and Circumradius

Median

The median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. The three medians intersect at the centroid of the triangle.

Inradius

The inradius is the radius of the largest circle that fits inside the triangle. It can be calculated as: inradius = Area / s, where s is the semiperimeter.

Circumradius

The circumradius is the radius of the circle that passes through all three vertices of the triangle. It can be calculated as: circumradius = a / (2 × sin(A)).