triangles

A plane triangle is an object having no thickness and 3 straight sides. (They are flat, or two-dimensional. Triangles in spaces with more dimensions, for example a triangle on the surface of a globe, are not described here.) Every triangle has 3 sides, 3 vertices (the points where the sides meet), and 3 angles.

Triangle calculators

If the sizes of three of the six parts (3 angles and 3 sides) of a triangle are known, and at least one of the known parts is a side, the sizes of the other sides and angles can be calculated. Mathematicians call this, “solving the triangle.”

In what follows, the sides are named with lowercase letters, and the angle opposite a side is named by the same letter, but in uppercase.

All three sides

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Two angles and the side between them

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Two sides and the angle between them

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Two sides and an opposite angle

Knowing two sides and the angle at the end of one of the sides is not enough information to determine a triangle uniquely. From the data, it may be possible to construct one triangle, two different triangles, or no triangle at all. Try it and see.

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This angle must be opposite side a.

 



triangle one



  triangle two
  
  
  
  
 



Equations about triangles

labeled drawing of triangle

For the equations below, the angles are named with the Greek letters alpha (α), beta (β) and gamma (γ).  Side a is opposite angle alpha, side b opposite angle beta, and side c opposite angle gamma.

The Law of Sines

equation as a graphic

The Cosine Law

equation as a graphic

The Law of Tangents

equation as a graphic

Area

drawing defining the altitude of a triangle

The most commonly encountered formula for a triangle's area is the geometric one: half the product of the length of a side (b in the drawing) and the length of a line (called the altitude, labeled h in the drawing) perpendicular to that side and drawn from the opposite vertex. Let K = the area of the triangle.  

equation as a graphic

equation as a graphic

The area can also be calculated from the lengths of the three sides alone (Heron's formula). Let s equal half the perimeter.

equation as a graphic

The inscribed circle

A circle can be inscribed in any triangle; its center is at the point where the lines bisecting the triangle’s angles meet.  Its radius is given by (s as defined above):

equation as a graphic

The half-angle formulas

Using s and r as defined above,

equation as a graphic

Right Triangles

The most famous of triangle equations, the Pythagorean Theorem:

c² = a² + b²

where c is the hypotenuse, the side opposite the right angle.

Isosceles Triangles

An isosceles triangle is one in which two sides are the same length.

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