TutorialKart.comWhat Is a Triangle?
A triangle is a closed two-dimensional shape formed by three straight line segments. It has exactly three sides, three vertices, and three interior angles.
The word triangle comes from tri, meaning three, and angle. A triangle can therefore be described as a polygon with three angles.
A triangle is commonly named using its vertices. For example, a triangle with vertices A, B, and C is written as △ABC. Its sides are AB, BC, and CA, while its angles are ∠A, ∠B, and ∠C.
Parts and Terminology of a Triangle
The following terms describe the main parts and important line segments associated with a triangle.
Vertices of a Triangle
A vertex is a point at which two sides of a triangle meet. The plural form of vertex is vertices.
In the figure, A, B, and C are the three vertices of the triangle. They are highlighted in blue.
Sides of a Triangle
A side is a line segment joining two vertices. Every triangle has three sides.
In the diagram, the sides are AB, BC, and CA.
Interior Angles of a Triangle
An interior angle is formed inside the triangle where two sides meet at a vertex. Every triangle has three interior angles.
In the diagram, the three interior angles are located at vertices A, B, and C.
Altitude of a Triangle
An altitude is a perpendicular line segment drawn from a vertex to the opposite side or to the line containing the opposite side. The perpendicular distance represented by an altitude is also called the height of the triangle.
In the figure, the altitude is drawn from vertex C perpendicular to the opposite side AB.
A triangle has three altitudes: one from each vertex. In an acute triangle, all three altitudes lie inside the triangle. In a right triangle, two sides themselves act as altitudes. In an obtuse triangle, two altitudes meet extensions of the opposite sides.
Angle Bisector of a Triangle
An angle bisector is a line segment or ray that passes through a vertex and divides the angle at that vertex into two equal angles.
In the figure, CP is the angle bisector of ∠C. Therefore, the two angles formed on either side of CP are equal.
Median of a Triangle
A median is a line segment joining a vertex to the midpoint of the opposite side. Every triangle has three medians, and they meet at a point called the centroid.
An altitude and a median are not generally the same line segment. They coincide in some special cases, such as the altitude drawn to the base of an isosceles triangle.
Types of Triangles Based on Side Lengths
Triangles can be classified by comparing the lengths of their three sides.
| Type of triangle | Side-length condition | Angle relationship |
| Equilateral triangle | All three sides are equal in length. | All three angles are equal, and each angle measures 60°. |
| Isosceles triangle | At least two sides are equal in length. | The angles opposite the equal sides are equal. |
| Scalene triangle | All three sides have different lengths. | All three angles have different measures. |
Under the common school convention, an isosceles triangle is described as having two equal sides. In a broader mathematical definition, an equilateral triangle may also be treated as a special isosceles triangle because it has at least two equal sides.
Types of Triangles Based on Angle Measures
Triangles can also be classified according to the measures of their interior angles.
| Type of triangle | Angle condition |
| Acute triangle | All three interior angles are less than 90°. |
| Right triangle | One interior angle is exactly 90°. |
| Obtuse triangle | One interior angle is greater than 90°. |
A triangle cannot have more than one right angle or more than one obtuse angle because the sum of its interior angles must be 180°.
Important Properties of a Triangle
The following properties apply to every valid triangle.
Triangle Interior Angle Sum
The sum of the three interior angles of a triangle is always 180°.
If the angles are represented by ∠A, ∠B, and ∠C, then:
∠A + ∠B + ∠C = 180°
For example, if two angles measure 50° and 60°, the third angle is 180° − 50° − 60° = 70°.
Triangle Inequality Property
The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
For side lengths a, b, and c, all three conditions must be true:
a + b > cb + c > ac + a > b
For example, lengths 3, 4, and 5 can form a triangle. Lengths 2, 3, and 5 cannot form a triangle because 2 + 3 is not greater than 5.
Relationship Between Sides and Opposite Angles
The longest side of a triangle lies opposite its largest angle. Similarly, the shortest side lies opposite its smallest angle. Equal sides lie opposite equal angles.
Exterior Angle Property
An exterior angle is formed when one side of a triangle is extended. Its measure equals the sum of the two non-adjacent interior angles.
For example, if the two remote interior angles measure 45° and 65°, the exterior angle measures 45° + 65° = 110°.
Circumcircle of a Triangle
Every triangle has a circle that passes through all three vertices. This circle is called the circumcircle, and its center is called the circumcenter.
The circumcenter is the point at which the perpendicular bisectors of the three sides meet.
Incircle of a Triangle
Every triangle has a circle that touches all three sides. This circle is called the incircle, and its center is called the incenter.
The incenter is the point at which the three internal angle bisectors meet.
Perimeter and Area of a Triangle
Triangle Perimeter Formula
The perimeter of a triangle is the total length of its three sides.
If the side lengths are a, b, and c, then:
Perimeter = a + b + c
For example, a triangle with side lengths 5 cm, 7 cm, and 8 cm has a perimeter of 5 + 7 + 8 = 20 cm.
Triangle Area Formula Using Base and Height
The area of a triangle is half the product of its base and corresponding perpendicular height.
Area = 1/2 × base × height
For example, if the base is 10 cm and the perpendicular height is 6 cm, the area is 1/2 × 10 × 6 = 30 cm².
How to Check Whether Three Lengths Form a Triangle
To determine whether three positive lengths can form a triangle, arrange them from smallest to largest. Then add the two smaller lengths. They form a triangle only when their sum is greater than the largest length.
- Confirm that all three lengths are positive.
- Identify the largest length.
- Add the other two lengths.
- Check whether their sum is greater than the largest length.
For lengths 6, 8, and 10, the check is 6 + 8 > 10. Since 14 > 10, the lengths form a triangle.
For lengths 4, 5, and 9, the check is 4 + 5 > 9. Since the sum equals 9 rather than exceeding it, the lengths do not form a triangle.
Frequently Asked Questions About Triangles
How many sides, vertices, and angles does a triangle have?
A triangle has three sides, three vertices, and three interior angles.
What is the sum of the angles in a triangle?
The sum of the three interior angles of every triangle is 180°.
Can a triangle have two right angles?
No. Two right angles would already total 180°, leaving no measure for the third angle. A valid triangle can have only one right angle.
What is the difference between an altitude and an angle bisector?
An altitude is perpendicular to the opposite side, while an angle bisector divides an angle into two equal parts. These line segments may coincide in special triangles but have different definitions.
Can three equal lengths form a triangle?
Yes. Three equal positive lengths form an equilateral triangle. Each interior angle of that triangle measures 60°.
Triangle Concepts Summary
A triangle is a three-sided polygon with three vertices and three interior angles. Triangles can be classified by their side lengths as equilateral, isosceles, or scalene, and by their angles as acute, right, or obtuse. Their interior angles total 180°, and their side lengths must satisfy the triangle inequality.
In this Geometry tutorial, we covered the parts of a triangle, types of triangles, important triangle properties, and the basic formulas for perimeter and area.