Triangle

Pronunciation: /ˈtraɪˌæŋ gəl/ ?

A triangle is a three sided polygon[2][3]. All triangles have three non-collinear sides made up of straight line segments. All triangles have three angles.

Examples of triangles Non-examples of triangles
Figure 1: Examples of triangles Figure 2: Examples of shapes that are not triangles.

Article Index

Types of Triangles
empty spaceParts of a Triangle
empty spaceRight Triangle
empty spaceAcute Triangle
empty spaceObtuse Triangle
empty spaceScalene Triangle
empty spaceEquilateral Triangle
empty spaceIsosceles Triangle
Labeling Triangles
Properties of Triangles
empty spacePerimeter of a Triangle
empty spaceAngle Sum Theorem
empty spaceArea of a Triangle
empty spaceHeron's Formula for Area of a Triangle
empty spaceIncircle and Incenter of a Triangle
empty spaceCircumcircle and Circumcenter of a Triangle
empty spaceMedian of a Triangle
empty spaceCentroid of a Triangle
empty spaceAltitude of a Triangle
empty spaceOrthocenter of a Triangle
empty spaceSAS Congruence
empty spaceEuclid. 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

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Manipulative 1: Parts of a Triangle. Created with GeoGebra.

A triangle has three angles, three vertices, three sides and 3 pairs of exterior angles.

Types of Triangles

ExampleName Click for more information.Description
right triangleRight triangleA triangle with one right angle.
acute triangleAcute triangleA triangle with three acute angles.
obtuse triangleObtuse triangleA triangle with one obtuse angle.
scalene triangleScalene triangleA triangle whose sides are all different lengths.
equilateral triangleEquilateral triangleA triangle with three equal sides.
Isosceles triangleIsosceles triangleA triangle with two equal sides.
Figure 3: Types of triangles

Labeling Triangles

A labeled triangle.
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
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Manipulative 1: 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. Click on the blue points in manipulative 1 and drag them to change the figure.
Angle Sum Theorem
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Manipulative 2: Angle Sum 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. Click on the blue points in manipulative 2 and drag them to change the figure.
Area
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Manipulative 3: Area of a triangle. Created with GeoGebra.
The area of a triangle is (1/2)*b*h 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 in manipulative 3 and drag them to change the figure.
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Manipulative 4: 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 s=(a+b+c)/2.

Heron's formula for the area of a triangle is
A = square root(s*(s-a)*(s-b)*(s-c)).
Incircle
Incenter
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Manipulative 5: 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 from All Math Words Encyclopedia.

Click on the blue points in the manipulative and drag them to change the figure.

Circumcircle
Circumcenter
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Manipulative 6: 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
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Manipulative 7: Triangle centroid. 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 triangle1. The centroid is at the point where the medians intersect.

Triangle Altitude
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Manipulative 8: Triangle altitude. 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
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Manipulative 9: Triangle orthocenter. Created with GeoGebra.
The orthocenter of a triangle is at the intersection of the altitudes of a triangle.
SAS Congruence
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Manipulative 10: Triangle SAS congruence. 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.

For more information on SAS Congruence, see SAS Congruence from All Math Words Encyclopedia.

Proposition 6, Euclid's Elements: If two angles of a triangle are equal, the sides opposite the equal angles are also equal.
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Manipulative 11: Euclid proposition 6 SAS congruence. 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.

For more information on this property of triangles see:

Centers of a Triangle

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Manipulative 12: Centers of a triangle. Created with GeoGebra.

check mark 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

More Information

  • McAdams, David. Angle. allmathwords.org. Life is a Story Problem LLC. 2009-03-12.

Cite this article as:


Triangle. 2010-06-30. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/t/triangle.html.

Translations

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Revision History


2010-06-30: Changed 'sum of angles of a triangle' to 'Angle Sum Theorem' in subtitles. (McAdams, David.)
2010-01-02: Added "References" (McAdams, David.)
2009-11-28: Added 'Parts of a Triangle'. (McAdams, David.)
2008-10-30: Changed all manipulatives to GeoGebra (McAdams, David.)
2008-09-15: Added wikipedia to more information (McAdams, David.)
2008-06-07: Corrected spelling errors (McAdams, David.)
2008-03-20: Clarified definition of scalene triangle (McAdams, David.)
2007-08-29: Added "Centers of a Triangle" (McAdams, David.)
2007-08-24: Expanded (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)

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