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