The area of ​​an equilateral triangle

Among the geometric figures thatare considered in the geometry section, most often you have to deal with solving certain problems with a triangle. It is a geometric figure formed by three lines. They do not intersect at one point and are not parallel. You can give a different definition: a triangle is a broken closed line, consisting of three links, where its beginning and end are connected at one point. If all three sides have the same value, then this is the right triangle, or, as they say, equilateral.

How to determine the area of ​​equilateraltriangle? To solve such problems it is necessary to know some properties of this geometric figure. First, for a given kind of triangle, all angles are equal. Secondly, the height that descends from the top to the bottom is simultaneously a median and a height. This indicates that the height divides the vertex of the triangle by two equal angles, and the opposite side into two equal segments. Since an equilateral triangle consists of two right-angled triangles, the Pythagorean theorem must be used to determine the desired value.

Calculation of the area of ​​the triangle can be done in various ways, depending on the known quantities.

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 of the side and height. In the form of a formula, this will look like this:

S = 1/2 * h * b

In words, the area of ​​an equilateral triangle is equal to one second of its side and height.

2. If only the magnitude of the side is known, then before calculating the area, it is necessary to calculate its height. To do this, consider half the triangle, in which the height will be one of the legs, the hypotenuse is the side of the triangle, and the second one is half the side of the triangle according to its properties. From the same Pythagorean theorem, we determine the height of the triangle. As it is known, 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 the hypotenuse, half of the side - one leg, and the height - the second.

(b / 2) ² + h2 = b², from here

h² = b²- (b / 2) ². We reduce to the common denominator:

h² = 3b² / 4,

h = √3b² / 4,

h = b / 2√3.

As we 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 the three.

3. There are also problems where it is necessary to determine the area of ​​an equilateral triangle at a certain height. And it turns out to be simple. We have already deduced in the previous case that h² = 3 b² / 4. Next it is necessary 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, which is the area, we get:

S = 1/2 * h * 2h / √3, hence S = h² / √3.

There are tasks when it is necessary to findthe area of ​​an equilateral triangle along the radius of the 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 act according to the principle we know. With a known radius, we derive a side from the formula and calculate it by substituting a known radius value. The obtained value is substituted into the already known formula for calculating the area of ​​the regular triangle, we perform arithmetic calculations and find the required value.

As we see, in order to solve similartask, you need to know not only the properties of the right triangle, but the Pythagorean theorem, and the radius of the circumscribed and inscribed circle. For those who know this solution of such problems will not be difficult.

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