# Area of a triangle

In 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, the application of which will depend on whether we know the lengths of its three sides (Heron's Formula) or two together with the angle they form.

## Definition

The area of a triangle it is the measure of the surface enclosed by the three sides of the triangle.

###  Classic formula

It is equal to the base times the height of the triangle divided by 2:

A = (b · h) / 2

Height is the segment perpendicular from one vertex to the line that contains the opposite side.

• Equilateral triangle: h = (√3 / 2) · l

A = (√3 · l²) / 4

• Right triangle: A = (b · c) / 2

That is, it is equal to the product of the legs divided by.

Example 1: Find the area of the triangle whose base measures 2 cm and its height 3 cm.

Making use of the triangle area formula:

A = (b · c) / 2 = (2 cm · 3 cm) / 2) = 6/2 = 3 cm²

Example 2: Find the area of the equilateral triangle whose sides measure 1.

Making use of the equilateral triangle area formula:

A = (√3 · l²) / 4 = [√3 · (1 cm) ²] / 4 = 0.43 cm²

### Heron's formula

This is attributed to the Greek mathematician, Heron of Alexandria. With it we can calculate the area of the triangle knowing the lengths of its three sides a, b and c.

A = √ [s (s - a) (s - b) (s - c)]

Where s is the half-perimeter (half of the perimeter, that is, half of the sum of the lengths of its sides) of the triangle:

s = (a + b + c) / 2

Example: Find the area of the triangle whose sides measure 5 cm, 3 cm, and 2 cm.

Let a = 5 cm, b = 3 cm and c = 2 cm; We calculate your semi-perimeter first:

s = (a + b + c) / 2 = (5 cm + 3 cm + 2 cm) / 2 =

= 10/2 = 5 cm

Now, applying Heron's formula A = √ [s (s - a) (s - b) (s - c)], we have:

A = √ [10 cm (10 cm - 5 cm) (10 cm - 3 cm) (10 cm - 2 cm)]

= √ [10 cm (5 cm) (7 cm) (8 cm)] =

= √ [10 cm (280 cm4)]

= √ (2800 cm5)

= 52.92 cm²

### Knowing two sides and the angle they form

If we know two sides and the angle they form, we can use the following formula:

A = [(b · a) sin C] / 2

Example: Calculate the area of the triangle whose two sides measure 12 cm and 20 cm and the angle they make is 60⁰.

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