Area of a triangle using trigonometry

triangleIn general, a triangle it is a three-sided geometric figure. These can be represented with a capital letter, the angles with a lowercase letter and the vertices with the same previous letters or with Greek letters.

To calculate its area, we can use the formula A = ½ (Base · Height). However, there are other procedures, among them is the trigonometric whose application will depend on whether we know the lengths of two of its sides next to the angle they form.

Definition

triangle areaThe area of a triangle it is the measure of the surface enclosed by the three sides of the triangle. To calculate it we generally use the formula:

A = (base · height) / 2

That is, the area is equal to the base times the height of the triangle divided by 2. Where the height "h" is the perpendicular segment from one vertex to the line that contains the opposite side.

Calculation of the area knowing two sides and the angle they form

area triangulo trigonometryThis is known as the trigonometry form; Suppose we have a triangle (like the one on the right) of which we know two sides and the angle they form. To calculate its area we use the formula A = (b · h) / 2, therefore, we must determine its height.

From the figure we obtain:

without C = h / a

Clearing h:

h = a · sin C

Substituting in the area formula:

A = [(b · a) sin C] / 2

Examples

1. Calculate the area of the triangle whose two sides measure 12 cm and 20 cm and the angle they make is 60 ⁰.

Let a = 12 cm, b = 20 cm and C = 60 ⁰, then:

A = [(b · a) without C] / 2 = [(20 cm · 12 cm) (without 60 ⁰)] / 2 =

   = (207.8 cm²) / 2 =

   = 103.9 cm²

2. Calculate the area of the triangle whose two sides measure 18 cm and 25 cm and the angle they make is 30 ⁰.

Let a = 18 cm, b = 25 cm and C = 30 ⁰, then:

A = [(b · a) without C] / 2 = [(25 cm · 10 cm) (without 30 ⁰)] / 2 =

   = (2250 cm²) / 2 =

   = 1125 cm²