Among the geometrical figures that are considered in the geometry section, one often encounters a triangle when solving various problems. It is a geometric figure formed by three straight lines. They do not intersect at one point and are not parallel. A different definition can be given: a triangle is a broken polygonal line consisting of three links, where its beginning and end are connected at one point. If all three sides are of equal size, then this is a regular triangle, or, as they say, equilateral.
How to determine the area of an equilateral triangle? To solve such problems, it is necessary to know some properties of this geometric figure. Firstly, in this type of triangle, all angles are equal. Secondly, the height that descends from the top to the bottom is both the median and the height. This suggests that the height divides the top of the triangle into two equal angles, and the opposite side into two equal segments. Since an equilateral triangle consists of two right-angled triangles, it is necessary to use the Pythagorean theorem to determine the desired quantity.
The calculation of the area of a triangle can be done in various ways, depending on the known values.
1. Consider an equilateral triangle with known side b and height h. The area of the triangle in this case will be equal to one second product of the side and height. In the form of a formula, it will look like this:
S = 1/2 * h * b
In words, the area of an equilateral triangle is equal to one second product of its side and height.
2. If only the size of the side is known, then before looking for the area, it is necessary to calculate its height. To do this, consider the half of the triangle in which the height will be one of the legs, the hypotenuse is the side of the triangle, and the second leg is half the side of the triangle according to its properties. All of the same Pythagorean theorem will determine the height of the triangle. As is known from it, the square of the hypotenuse corresponds to the sum of the squares of the legs. If we consider half the triangle, then in this case the side is hypotenuse, half of the side is one leg, and the height is the second.
(b / 2) ² + h2 = b², hence
h² = b²- (b / 2) ². We bring to a common denominator:
h² = 3b² / 4,
h = √3b² / 4,
h = b / 2√3.
As you can see, the height of the figure in question is equal to the product of half of its side and the root of three.
Substitute in the formula and see: S = 1/2 / b * b / 2√3 = b² / 4√3.
That is, the area of an equilateral triangle is equal to the product of the fourth part of the square of the side and the root of three.
3. There are also such problems where it is necessary to determine the area of an equilateral triangle at a known height. And it turns out to be simple. We have already deduced in the previous case that h² = 3 b² / 4. Next, you need to get the side out of here and substitute it into the area formula. It will look like this:
b² = 4/3 * h², hence b = 2h / √3. Substituting in the formula by which the area is, we get:
S = 1/2 * h * 2h / √3, hence S = h² / √3.
There are problems when it is necessary to find the area of an equilateral triangle by the radius of an inscribed or circumscribed circle. For this calculation, there are also certain formulas that look like this: r = √3 * b / 6, R = √3 * b / 3.
We are acting on a principle familiar to us. With a known radius, we derive the side from the formula and calculate it by substituting a known radius value. The resulting value is substituted into the already known formula for calculating the area of a regular triangle, we carry out arithmetic calculations and find the desired value.
As we see, in order to solve similar problems, it is necessary to know not only the properties of the regular triangle, but also the Pythagorean theorem, and the radius of the circumscribed and inscribed circles. For those who have this knowledge, solving such problems will not be particularly difficult.