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triangle (Definition)

A triangle is a bounded planar region delimited by 3 straight lines.

\includegraphics{triangulo}

In Euclidean geometry, the angle sum of a triangle is always equal to $ 180^\circ$. In the figure: $ A+B+C=180^\circ$.

In hyperbolic geometry, the angle sum of a triangle is always strictly positive and strictly less than $ 180^\circ$. In the figure: $ 0^\circ<A+B+C<180^\circ$.

In spherical geometry, the angle sum of a triangle is always strictly greater than $ 180^\circ$ and strictly less than $ 540^\circ$. In the figure: $ 180^\circ<A+B+C<540^\circ$.

Also in spherical geometry, a triangle has these additional requirements: It must be strictly contained in a hemisphere of the sphere that is serving as the model for spherical geometry, and all of its angles must have a measure strictly less that $ 180^{\circ}$.

Triangles can be classified according to the number of their equal sides. So, a triangle with 3 equal sides is called equilateral, a triangle with 2 equal sides is called isosceles, and finally a triangle with no equal sides is called scalene. Notice that an equilateral triangle is also isosceles, but there are isosceles triangles that are not equilateral.

\includegraphics{trianglebyside}

In Euclidean geometry, triangles can also be classified according to the size of the greatest of its three (inner) angles. If the greatest of these is acute (and therefore all three are acute), the triangle is called an acute triangle. If the triangle has a right angle, it is a right triangle. If the triangle has an obtuse angle, it is an obtuse triangle.

\includegraphics{trianglebyangle}

Area of a triangle

There are several ways to calculate a triangle's area.

In hyperbolic and spherical geometry, the area of a triangle is equal to its defect (measured in radians).

For the rest of this entry, only Euclidean geometry will be considered.

Many formulas for the area of a triangle exist. The most basic one is $ \displaystyle A=\frac{1}{2}bh$, where $ b$ is its base and $ h$ is its height. Following is a derivation of another formula for the area of a triangle.

Let $ a,b,c$ be the sides and $ A,B,C$ the interior angles opposite to them. Let $ h_a,h_b,h_c $ be the heights drawn upon $ a,b,c$ respectively, $ r$ the inradius and $ R$ the circumradius. Finally, let $ \displaystyle s=\frac{a+b+c}{2}$ be the semiperimeter. Then


Area $\displaystyle =$ $\displaystyle \frac{a h_a}{2}=\frac{b h_b}{2}=\frac{c h_c}{2}$  
  $\displaystyle =$ $\displaystyle \frac{ab\sin C}{2}=\frac{bc\sin A}{2}=\frac{ca\sin B}{2}$  
  $\displaystyle =$ $\displaystyle \frac{abc}{4R}$  
  $\displaystyle =$ $\displaystyle sr$  
  $\displaystyle =$ $\displaystyle \sqrt{s(s-a)(s-b)(s-c)}$  

The last formula is known as Heron's formula.

Inequalities for the area are Weizenbock's inequality and the Hadwiger-Finsler inequality.

Angles in a triangle

  1. the sum of the angles in a triangle is $ \pi$ radians ($ 180^\circ$)
  2. sines law
  3. cosines law
  4. Mollweide's equations

Special geometric objects for a triangle

  1. incenter
  2. inscribed circle
  3. circumcenter
  4. circumscribed circle
  5. centroid
  6. orthocenter
  7. Lemoine point, Lemoine circle
  8. Gergonne point, Gergonne triangle
  9. orthic triangle
  10. pedal triangle
  11. medial triangle
  12. Euler Line



"triangle" is owned by Wkbj79. [ full author list (3) | owner history (2) ]
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See Also: sines law, Euler line, median, Pythagorean theorem, hypotenuse, orthocenter, orthic triangle, isosceles triangle, Ceva's theorem, cevian, sines law proof, fundamental theorem on isogonal lines, incenter, equilateral triangle, trigonometric version of Ceva's theorem, Heron's formula, Lemoine circle, circumcircle, Simson's line, Morley's theorem, Apollonius theorem, scalene triangle, centroid, equivalent conditions for triangles, equiangular triangle, regular triangle, pentagon, height, incircle, incircle radius determined by Pythagorean triple

Also defines:  acute triangle, right triangle, obtuse triangle
Keywords:  Geometry, Polygon, Angle

Attachments:
some proofs for triangle theorems (Proof) by Wkbj79
when is a point inside a triangle (Theorem) by matte
triangle solving (Definition) by stevecheng
limiting triangle (Definition) by Wkbj79
ideal triangle (Definition) by Wkbj79
triangle mid-segment theorem (Theorem) by pahio
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Cross-references: Euler line, medial triangle, pedal triangle, orthic triangle, Gergonne triangle, Gergonne point, Lemoine circle, Lemoine point, orthocenter, centroid, circumscribed, circumcenter, circle, inscribed, incenter, Mollweide's equations, cosines law, sines law, sum, Hadwiger-Finsler inequality, Weizenbock's inequality, inequalities, Heron's formula, semiperimeter, circumradius, inradius, interior angles, height, base, radians, defect, area of a triangle, area, obtuse angle, right angle, acute, inner, equilateral, scalene, isosceles, sides, measure, angles, sphere, contained, spherical geometry, positive, strictly, hyperbolic geometry, angle sum, Euclidean geometry, lines, region, planar
There are 231 references to this entry.

This is version 37 of triangle, born on 2001-10-06, modified 2007-06-18.
Object id is 139, canonical name is Triangle.
Accessed 41387 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
 51M05 (Geometry :: Real and complex geometry :: Euclidean geometries and generalizations)
 51M10 (Geometry :: Real and complex geometry :: Hyperbolic and elliptic geometries and generalizations)
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