Triangle

For other uses, see Triangle (disambiguation).

A black triangle

A triangle is a shape, or a part of two dimensional space. It has three straight sides and three vertices. The three angles of a triangle always add up to 180° (180 degrees). It is the polygon with the least possible number of sides. A triangle with vertices A, B, C is written as [math]\displaystyle{ \triangle ABC }[/math]. The study of geometry related to triangles is called trigonometry. Modern computers usually use triangles to make more complex graphic images or shapes.

Types of triangles

This is an isosceles right triangle, because it has a right angle and two of its sides have the same length.

Triangles can be grouped according to how many of their sides are equal:

• if all the three sides of a triangle have the same length, then it is an equilateral triangle.
• if a triangle has two sides with the same length, then it is an isosceles triangle.
• if all the three sides of a triangle have different lengths, then we have a scalene triangle.

Triangles can also be grouped by their angles:

• if a triangle has a right angle, that is, if one of the angles of that triangle measures 90° (90 degrees), then it is a right triangle. The opposite side to the right angle is the hypotenuse.
• if a triangle has an obtuse angle, that is, if one of the angles of that triangle is larger than 90°, then it is an obtuse triangle.
• if a triangle has only acute angles, that is, if all the angles of that triangle are less than 90°, then it is an acute triangle.

Equilateral triangle Isosceles triangle Scalene triangle Right Angled triangle Obtuse Angled triangle Acute Angled triangle

Area

b is base and h is height

The area of a triangle is equal to base times height times one half.

[math]\displaystyle{ area= \frac{base \times height}{2} }[/math]

Triangle Media

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  The circumcenter is the center of a circle passing through the three vertices of the triangle; the intersection of the altitudes is the orthocenter. The intersection of the angle bisectors is the center of the incircle.

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  The measures of the interior angles of the triangle always add up to 180 degrees (same color to point out they are equal).

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  The area formula for a triangle can be proven by cutting two copies of the triangle into pieces and rearranging them into a rectangle.

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  Applying trigonometry to find the altitude h

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  Orange triangles △ABC share a base AB and area. The locus of their apex C is a line (dashed green) parallel to the base. This is the Euclidean version of Lexell's theorem.

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  Rigidity of a triangle and square

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  Triangulation in a simple polygon

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  The Lemoine hexagon inscribed in a triangle

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  Circular triangles with a mixture of convex and concave edges

Related pages

• Law of sines
• Pythagorean theorem
• Sierpinski triangle
• Reuleaux triangle