include geometric shapes, which are discussed in the section geometry, the most frequently encountered in solving various problems of the triangle.It is a geometric figure formed by three lines.They do not intersect the same point and are not parallel.You can give another definition: a triangle is a broken closed line consisting of three units, where its beginning and end are connected at one point.If all three sides have the same value, then it is an equilateral triangle, or as they say, is equilateral.
How do we determine the area of an equilateral triangle?To solve these problems it is necessary to know some of the properties of geometric figures.Firstly, in the form of a triangle all angles are equal.Secondly, the height of which is lowered from the top of the base, is also the median, and high.This suggests that the height divides the apex of the triangle into two equal angles, and the opposite side - into two equal segments.Since equilateral triangle consists of two right-angled triangles, in determining the required quantity necessary to use the Pythagorean theorem.
Calculation of the area of a triangle can be made in different ways, depending on the known quantities.
1. Consider an equilateral triangle with the known side b, and height h.The area of the triangle in this case is equal to half the product side and height.In a formula would look like this:
S = 1/2 * h * b
the words, the area of an equilateral triangle is equal to half the product of its sides and height.
2. If you know only the value side, before seeking the area, it is necessary to calculate its height.For this we consider half of the triangle, which is the height of one of the legs, the hypotenuse - this side of the triangle, and the second leg - half of the triangle according to its properties.All the same Pythagorean theorem define the height of the triangle.As it is known from the square of the hypotenuse corresponds to the sum of the squares of the legs.If we consider half of the triangle, in this case, it is the hypotenuse side, half side - one the leg, and height - the second.
(b / 2) ² + h2 = b², here
h² = b²- (b / 2) ².Here is a common denominator:
h² = 3b² / 4,
h = √3b² / 4,
h = b / 2√3.
As you can see, the height of the figure under consideration is equal to the half of his face and 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 fourth part of the square root of the parties and of the three.
3. There are some tasks where you need to determine the area of an equilateral triangle at a certain height.And it is easier than ever.We have already brought in the previous case that h² = 3 b² / 4.Next you need to withdraw from this side and substitute in the area.It will look like this:
b² = 4/3 * h², hence b = 2h / √3.Substituting in the formula for which is an area we obtain:
S = 1/2 * h * 2h / √3, hence S = h² / √3.
We have the problem when you need to find the area of an equilateral triangle, the radius of the inscribed or circumscribed circle.For this calculation, there are also certain formula, which are as follows: r = b * √3 / 6, R * b = √3 / 3.
We act already familiar to us on principle.At a certain radius, we deduce from the formula and calculate its side, substituting the known value of the radius.The resulting value is substituted into the already well-known formula for calculating the area of an equilateral triangle, perform arithmetic calculations and find the desired value.
As you can see, in order to solve similar problems, you need to know not only the properties of an equilateral triangle and and the Pythagorean theorem, and the radius of the inscribed circle and.To possess this knowledge to resolve such problems will not pose much difficulty.
