An equilateral triangle is a triangle with three sides of equal length, it is a special isosceles triangle: three sides of equal length, three angles are equal and each angle is 60 degrees, with symmetry and stability, the equilateral triangle can be divided into two isosceles triangles to derive its area formula. There are two methods for calculating the area formula of equilateral triangles:

1, side length calculation method (only know the side length) :

The formula for the area of an equilateral triangle is: the square of the length of the side multiplied by the square root of three divided by four, so S = (a^2 * √3) / 4, where a is the length of the side of the equilateral triangle.

2, altimeter algorithm (knowing only the height):

Altimeter algorithm, the area of an equilateral triangle is equal to 3 times the height squared times the square root of 3 divided by 4. The formula is S = (3h^2√3)/4. Where S is the area, h is the height and √3 is the square root of 3.

Since the height of an equilateral triangle is also the length of the central line and the length of the high line, the height of an equilateral triangle can be calculated by dividing it into two isosceles triangles.

The height, centre line, angle bisector, vertical centre, external centre, internal centre, centre of gravity and other special points of an equilateral triangle coincide with the centre of gravity of the triangle.

To calculate the area, the side length calculation method is more common, and the side length calculation formula can be derived by dividing the equilateral triangle into two isosceles triangles.

Real life applications:

1. An equilateral triangle with a side length of 6 metres and an area of (6^2 * √3) / 4 = 9√3 square metres.

2. An equilateral triangle with a height of 5.2 cm can be calculated using the formula S = (3h^2√3)/4. Substituting the height into the formula gives S = (3 × 5.2^2 × √3)/4 ≈ 56.9 square centimetres.

3. An equilateral triangular pool with a side length of 10 metres and an area of (10^2 * √3)/4 = 25√3 square metres.

You need to pay attention to the knowledge point: “triangle area formula”, “right triangle area formula”, “equilateral triangle area formula”, “hyperbolic centred triangle area formula”, “elliptic centred triangle area formula”.

Equilateral triangles are symmetrical and stable, so they are often used in fields such as architecture, engineering and design.