2. Plane triangle formulae

A plane triangle, Figure C-2, has six parts: three angles and three sides.


Figure C-2
Plane triangle

 Individual angles may be acute (<90°), right (=90°), or obtuse (>90°) with their sum exactly 180°.

To geometrically define a triangle requires three parts, including at least one side, be fixed. Why a side? Because fixing angles alone does not constrain the size of the triangle. The two triangles in Figure C-3 are different sizes even though they have identical angles.

Figure C-3
Same angles; different triangle sizes

 Equations for triangle trigonometry are:

Angle condition: 


Equation C-1

Law of Sines: 


Equation C-2

Law of Cosines: 


Equation C-3

A right triangle is a special case where one angle is exactly 90°. Applying the Law of Cosines, Equation C-3, to a right triangle, Figure C-4:

Figure C-4
Right triangle

Because cos(90°) = 0

The Law of Cosines becomes the Pythagorean Theorem.

Depending on which parts of it are known the area of a triangle can be determined using one of two equations:

Using two sides and an included angle:


Equation C-4

Using three sides:


Equation C-5

where:      herro Equation C-6