# Triangle

Pronunciation: /ˈtraɪˌæŋ.gəl/ Explain

A triangle is a three sided polygon. All triangles have three non-collinear sides made up of straight line segments. All triangles have three angles.  Figure 1: Examples of triangles Figure 2: Examples of shapes that are not triangles.

### Article Index

Parts of a Triangle
Types of Triangles Right Triangle Acute Triangle Obtuse Triangle Scalene Triangle Equilateral Triangle Isosceles Triangle
Labeling Triangles
Properties of Triangles Perimeter of a Triangle Angle Sum Theorem Area of a Triangle Heron's Formula for Area of a Triangle Incircle and Incenter of a Triangle Circumcircle and Circumcenter of a Triangle Median of a Triangle Centroid of a Triangle Altitude of a Triangle Orthocenter of a Triangle SAS Congruence Euclid. Elements, Book 1 Proposition 6: If two sides of a triangle are equal, the angles opposite the equal sides are equal.
Centers of a Triangle

### Parts of a Triangle

 Click on the blue point and drag them to change the figure. Click on the check boxes to show or hide various parts Manipulative 1 - Parts of a Triangle Created with GeoGebra.

A triangle has three angles, three vertices, three sides and three pairs of exterior angles.

### Types of Triangles

 Example Name Click for more information. Description Right triangle A triangle with one right angle. Acute triangle A triangle with three acute angles. Obtuse triangle A triangle with one obtuse angle. Scalene triangle A triangle whose sides are all different lengths. Equilateral triangle A triangle with three equal sides. Isosceles triangle A triangle with two equal sides. Figure 3: Types of triangles

### Labeling Triangles Figure 4: Labeling triangles
By convention, triangles are usually labeled in a counterclockwise direction, often using the letters A, B, and C. The sides are often labeled with a lower case letter corresponding to the vertex opposite the side.

### Properties of Triangles

Perimeter
 Click on the blue points and drag them to change the figure. Manipulative 2 - Perimeter of a Triangle Created with GeoGebra.

The perimeter of a triangle is the sides of the triangle or the sum of the lengths of the sides. For example, if the lengths of the sides are 3, 4, and 5, the perimeter is 3 + 4 + 5 = 12.

Angle Sum Theorem
 Click on the blue points and drag them to change the figure. Can you change the triangle so that the sum of the interior angles is not equal to 180 degrees? Manipulative 3 - Triangle Sum of Angles Theorem Created with GeoGebra.
In Euclidean geometry, the sum of the angles of a triangle is 180° = π radians. In other geometries, this might not be true.
Area
 Click on the blue points and drag them to change the figure. Manipulative 4 - Area of a Triangle Created with GeoGebra.

The area of a triangle is where b (base) is any side of the triangle, and h (height) is the distance from the vertex opposite the base (in this case B) to the extended base (in this case the line AC).

 Click on the blue points and drag them to change the figure. Manipulative 5 - Heron's Formula for the Area of a Triangle Created with GeoGebra.

The area of a triangle can also be calculated from the length of the three sides using Heron's Formula. First, one must calculate the semiperimeter. This 1/2 of the perimeter. Since the perimeter is a + b + c where a, b and c are the length of the sides of the triangle, the semiperimeter is .

Heron's formula for the area of a triangle is .

Incircle
Incenter
 Click on the blue points and drag them to change the figure. Can the incenter ever be outside the triangle? Why? Manipulative 6 - Incircle and Incenter of a Triangle Created with GeoGebra.

The incircle of a triangle is the circle that is tangent to each of the sides of a triangle. The incenter is the center of the incircle. For more information on the incenter of a triangle, see Incenter.

Circumcircle
Circumcenter
 Click on the blue points and drag them to change the figure. Under what conditions is the circumcenter inide the triangle, on the edge of the triangle, outside the triangle? Manipulative 7 - Circumcircle and Circumcenter of a Triangle Created with GeoGebra.
The circumcircle of a triangle is the circle that passes through all of the vertices of a triangle. The circumcenter is the center of the circumcircle. For more information on the circumcenter or circumcircle of a triangle, see Circumcenter from All Math Words Encyclopedia.
Triangle Median
Triangle Centroid
 Click on the blue points and drag them to change the figure. Can the centroid ever be outside the triangle? Why or why not? Manipulative 8 - Centroid of a Triangle Created with GeoGebra.

A median of a triangle is a line drawn through a vertex of the triangle and the midpoint of the opposite side. This means that every triangle has three medians. The medians of a triangle meet at a point called the centroid of the triangle.

The centroid of a triangle is the center of gravity of the triangle. This means that if a triangle is balanced on a pin at the centroid, it would be perfectly balanced.

The centroid of a triangle is found by drawing two medians of the triangle. The centroid is at the point where the medians intersect.

Triangle Altitude
 Click on the blue points and drag them to change the figure. Under what conditions is the altitude outside of the triangle? Manipulative 9 - Altitude of a Side of a Triangle Created with GeoGebra.
An altitude of a triangle is a line segment from a vertex of the triangle to the extended opposite side, perpendicular to the opposite side.
Triangle Orthocenter
 Click on the blue points and drag them to change the figure. Under what conditions is the orthocenter inside the triangle, on the edge of the triangle, outside the triangle. Manipulative 10 - Orthocenter of a Triangle Created with GeoGebra.
The orthocenter of a triangle is at the intersection of the altitudes of a triangle.
SAS Congruence
 Click on the blue points and drag them to change the figure. Is there any way you can change these triangles so that they are not congruent? Manipulative 11 - Side-Angle-Side Congruence of Triangles Created with GeoGebra.

Two triangles are congruent if two adjacent sides and the angle contained by the sides are congruent with corresponding sides and angle of the other triangle. In this case we say that the triangles are SAS congruent. SAS stands for side, angle, side.

Proposition 6, Euclid's Elements: If two angles of a triangle are equal, the sides opposite the equal angles are also equal.
 Click on the blue points and drag them to change the figure. Manipulative 12 - Isosceles Triangle Created with GeoGebra.

In a triangle, if two angles have equal length, the sides opposite the equal angles are also equal. In figure 16, the angle ABC is equal to the angle ACB. The side AB is also equal to the side AC.

### Centers of a Triangle

 Click on the blue points and drag them to change the figure. For what class of triangle are the centroid, orthocenter and circumcenter coincidental? Manipulative 13 - Centers of a Triangle Created with GeoGebra.

#### Understanding Check

Write your answer on a piece of paper, then use your mouse to click on the 'Click for Answer' text to see the correct answer. Click on the yellow points and drag them to change the manipulative

1. For what type of triangle are the five centers shown the same point? Click for Answer
2. Which centers are always inside a triangle? Click for Answer
3. Which centers can be inside or outside a triangle? Click for Answer
4. Which center is on the hypotenuse of a right triangle? Click for Answer
5. Which center is on the vertex opposite the hypotenuse of a right triangle? Click for Answer
6. Which centers are always collinear (on the same line)? Click for Answer
7. For what type of triangle is the orthocenter inside the triangle? Click for Answer Outside the triangle? Click for Answer
1. McAdams, David E.. All Math Words Dictionary, triangle. 2nd Classroom edition 20150108-4799968. pg 181. Life is a Story Problem LLC. January 8, 2015. Buy the book

• McAdams, David E.. Angle. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 3/12/2009. http://www.allmathwords.org/en/a/angle.html.
• Euclid of Alexandria. Elements. Clark University. 9/6/2018. https://mathcs.clarku.edu/~djoyce/elements/elements.html.
• Wilson, Jim. Construction of the Nine-Point Circle. jwilson.coe.uga.edu. Jim Wilson's Home Page. University of Georgia. 12/16/2018. http://jwilson.coe.uga.edu/emt668/emt668.folders.f97/anderson/geometry/geometry1project/construction/construction.html.

McAdams, David E. Triangle. 5/13/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/t/triangle.html.

### Revision History

5/13/2019: Changed equations and expressions to new format. (McAdams, David E.)
4/3/2019: Corrected orthocenter manipulative. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
12/16/2018: Removed broken links, updated license, implemented new markup. Changed geogebra to new format. (McAdams, David E.)
6/30/2010: Changed 'sum of angles of a triangle' to 'Angle Sum Theorem' in subtitles. (McAdams, David E.)