The side of an equilateral triangle is 3. find its area.

The area of ​​any triangle is half the product of the base and the height. Let’s write the formula:

S = 1/2 × (a × h),

where S is the area of ​​the triangle, a is the base, h is the height.

Let us derive a formula for calculating the area of ​​an equilateral triangle.
The property of the height of an equilateral triangle: “In an equilateral triangle, the height drawn to either side is also its bisector and median.”

Let’s designate one of the sides of our equilateral triangle by the letter a.

Then, based on the property of the median, AH = HB = a / 2.

This means that the height BH divides our equilateral triangle into two right-angled ones. Let’s consider one of them. Let it be the ACH triangle.

The hypotenuse of our right-angled triangle is AC, one leg is CH, and the other is AH.

Using the Pythagorean theorem, we write down the expression for a given right-angled triangle and find the height CH = h. Pythagorean theorem: “The square of the hypotenuse is equal to the sum of the squares of the legs.”

(a / 2) ² + h² = a²;

h² = a² – (a / 2) ² = a² – a² / 4 = (4a² – a²) / 4 = 3a² / 4;

h = √ (3 a² / 4) = (a √ 3) / 2.

Substituting the value of the height h in the formula for the area of ​​a triangle, we get the formula for calculating the area of ​​an equilateral triangle:

S = 1/2 × (a × (a √3) / 2) = 1/2 × (a² √ 3) / 2 = (a² √ 3) / 4.

Calculate the area of ​​this equilateral triangle
S = (a² √ 3) / 4 = (3² √ 3) / 4 = (9 √ 3) / 4 = 3.9 sq. units

Answer: the area of ​​a given equilateral triangle is 3.9 square meters. units