Triangle Definition And Properties - Math Open Reference

Math Open Reference Home Contact About Subject Index Triangle A closed figure consisting of three line segments linked end-to-end. A 3-sided polygon. Try this Drag the orange dots on each vertex to reshape the triangle.

Triangle properties

Vertex	The vertex (plural: vertices) is a corner of the triangle. Every triangle has three vertices.
Base	The base of a triangle can be any one of the three sides, usually the one drawn at the bottom. You can pick any side you like to be the base. Commonly used as a reference side for calculating the area of the triangle. In an isosceles triangle, the base is usually taken to be the unequal side.
Altitude	The altitude of a triangle is the perpendicular from the base to the opposite vertex. (The base may need to be extended). Since there are three possible bases, there are also three possible altitudes. The three altitudes intersect at a single point, called the orthocenter of the triangle. See Orthocenter of a Triangle. In the figure above, you can see one possible base and its corresponding altitude displayed.
Median	The median of a triangle is a line from a vertex to the midpoint of the opposite side. The three medians intersect at a single point, called the centroid of the triangle. See Centroid of a Triangle
Area	See area of the triangle and Heron's formula
Perimeter	The distance around the triangle. The sum of its sides. See Perimeter of a Triangle
Interior angles	The three angles on the inside of the triangle at each vertex. See Interior angles of a triangle
Exterior angles	The angle between a side of a triangle and the extension of an adjacent side. See Exterior angles of a triangle

Also:

1. The shortest side is always opposite the smallest interior angle
2. The longest side is always opposite the largest interior angle

For more on this see Side / angle relationship in a triangle

Terminology

It is usual to name each vertex of a triangle with a single capital (upper-case) letter. The sides can be named with a single small (lower case) letter, and named after the opposite angle. So in the figure above, you can see that side b is opposite vertex B, side c is opposite vertex C and so on.

Alternatively, the side of a triangle can be thought of as a line segment joining two vertices. So then side b would be called AC. This is the form used on this site because it is consistent across all shapes, not just triangles.

Properties of all triangles

These are some well known properties of all triangles. See the section below for a complete list

• The interior angles of a triangle always add up to 180°
• The exterior angles of a triangle always add up to 360°

Types of Triangle

There are seven types of triangle, listed below. Note that a given triangle can be more than one type at the same time. For example, a scalene triangle (no sides the same length) can have one interior angle 90°, making it also a right triangle. This would be called a "right scalene triangle".

Isosceles	Two sides equal See Isosceles triangle definition
Equilateral	All sides equal See Equilateral triangle definition
Scalene	No sides equal See Scalene triangle definition

Right Triangle	One angle 90°. See Right triangle definition
Obtuse	One angle greater than 90° See Obtuse triangle definition
Acute	All angles less than 90° See Acute triangle definition
Equiangular	All interior angles equal See Equiangular triangle definition

Classifying triangles

The seven types of triangle can be classified two ways: by sides and by interior angles. For more on this see Classifying triangles.

Constructing triangles

Many types of triangle can be constructed using a a compass and straightedge using the traditional Euclidean construction methods. For more on this see Constructions using Compass and straightedge.

Other triangle topics

General

• Triangle definition
• Hypotenuse
• Triangle interior angles
• Triangle exterior angles
• Triangle exterior angle theorem
• Pythagorean Theorem
• Proof of the Pythagorean Theorem
• Pythagorean triples
• Triangle circumcircle
• Triangle incircle
• Triangle medians
• Triangle altitudes
• Midsegment of a triangle
• Triangle inequality
• Side / angle relationship

Perimeter / Area

• Perimeter of a triangle
• Area of a triangle
• Heron's formula
• Area of an equilateral triangle
• Area by the "side angle side" method
• Area of a triangle with fixed perimeter

Triangle types

• Right triangle
• Isosceles triangle
• Scalene triangle
• Equilateral triangle
• Equiangular triangle
• Obtuse triangle
• Acute triangle
• 3-4-5 triangle
• 30-60-90 triangle
• 45-45-90 triangle

Triangle centers

• Incenter of a triangle
• Circumcenter of a triangle
• Centroid of a triangle
• Orthocenter of a triangle
• Euler line

Congruence and Similarity

• Congruent triangles

Solving triangles

• Solving the Triangle
• Law of sines
• Law of cosines

Triangle quizzes and exercises

• Triangle type quiz
• Ball Box problem
• How Many Triangles?
• Satellite Orbits

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