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In the school geometry course, a huge amount of time is devoted to the study of triangles. Students calculate angles, build bisectors and heights, find out how shapes differ from each other, and the easiest way to find their area and perimeter. It seems that this is not useful in any way in life, but sometimes it is still useful to know, for example, how to determine that a triangle is equilateral or obtuse. How to do it?
Triangle types
Three points that do not lie on the same straight line, and the line segments that connect them. It seems that this figure is the simplest. What can triangles look like if they only have three sides? In fact, there are a fairly large number of options, and some of them are given special attention as part of the school geometry course. An equilateral triangle is an equilateral one, that is, all its angles and sides are equal. It has a number of remarkable properties, which will be discussed later.
The isosceles has only two equal sides, and it is also quite interesting. In a rectangular one, and as you might guess, one of the corners is straight or obtuse, respectively. However, they can also be isosceles.
There is also a special one called Egyptian. Its sides are 3, 4 and 5 units. However, it is rectangular. It is believed that it was actively used by Egyptian surveyors and architects to build right angles. It is believed that the famous pyramids were built with its help.
And yet all the vertices of a triangle can lie on one straight line. In this case, it will be called degenerate, while all the others are called non-degenerate. They are one of the subjects of study of geometry.
Triangle is equilateral
Of course, the correct figures are always of the greatest interest. They seem more perfect, more graceful. The formulas for calculating their characteristics are often simpler and shorter than for ordinary figures. This also applies to triangles. It is not surprising that a lot of attention is paid to them when studying geometry: schoolchildren are taught to distinguish regular figures from the rest, and they are also told about some of their interesting characteristics.
Features and properties
As the name suggests, each side of an equilateral triangle is equal to the other two. In addition, it has a number of features, thanks to which it is possible to determine whether the figure is correct or not.
If at least one of the above signs is observed, then the triangle is equilateral. For a regular figure, all the above statements are true.
All triangles have a number of remarkable properties. Firstly, the middle line, that is, the segment dividing the two sides in half and parallel to the third, is equal to half the base. Secondly, the sum of all the angles of this figure is always equal to 180 degrees. In addition, there is another interesting relationship in triangles. So, opposite the larger side lies a larger angle and vice versa. But this, of course, has nothing to do with an equilateral triangle, because all its angles are equal.
Inscribed and circumscribed circles
Often in a geometry course, students also learn how shapes can interact with each other. In particular, circles inscribed in polygons or described around them are studied. What is this about?
An inscribed circle is a circle for which all sides of the polygon are tangent. Described - the one that has points of contact with all corners. For each triangle, it is always possible to construct both the first and second circles, but only one of each type. The evidence for these two
theorems are given in the school course of geometry.
In addition to calculating the parameters of the triangles themselves, some tasks also involve calculating the radii of these circles. And the formulas for
equilateral triangle look like this:
where r is the radius of the inscribed circle, R is the radius of the circumscribed circle, a is the length of the side of the triangle.
Height, perimeter and area calculation
The main parameters that schoolchildren are involved in calculating while studying geometry remain unchanged for almost any figure. These are the perimeter, area and height. For ease of calculation, there are various formulas.
So, the perimeter, that is, the length of all sides, is calculated in the following ways:
P = 3a = 3√ ̅3R = 6√ ̅3r, where a is the side of a regular triangle, R is the radius of the circumscribed circle, r is the inscribed one.
h = (√ ̅3/2)*a, where a is the length of the side.
Finally, the formula is derived from the standard, that is, the product of half the base and its height.
S = (√ ̅3/4)*a 2 , where a is the length of the side.
Also, this value can be calculated through the parameters of the circumscribed or inscribed circle. There are also special formulas for this:
S = 3√ ̅3r 2 = (3√ ̅3/4)*R 2 , where r and R are the radii of the inscribed and circumscribed circles, respectively.
Building
Another interesting type of task, including triangles, is associated with the need to draw a particular shape using a minimal set
tools: a compass and a ruler without divisions.
In order to build a regular triangle with only these tools, you need to follow a few steps.
- It is necessary to draw a circle with any radius and with a center at an arbitrary point A. It must be noted.
- Next, you need to draw a straight line through this point.
- The intersections of the circle and the straight line must be designated as B and C. All constructions must be carried out with the greatest possible accuracy.
- Next, you need to build another circle with the same radius and center at point C or an arc with the appropriate parameters. Intersections will be marked D and F.
- Points B, F, D must be connected by segments. An equilateral triangle is built.
Solving such problems is usually a problem for schoolchildren, but this skill can be useful in everyday life.
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You can find the area of an equilateral triangle using any formula for an arbitrary figure of a given type, or use those that already take into account the peculiarity of this particular figure and the mathematical expressions are greatly simplified.
The first case only requires replacing all sides with the same value and taking into account that all angles of the triangle are 60º. Then it remains to carry out simple transformations, which will lead to the formulas given in finished form a little lower.
Formula 1: known side
In this and subsequent formulas, the standard notation for the magnitudes of a triangle is adopted. More details can be found in the table below.
The calculation of the area of a triangle in this case will be carried out according to the formula:
S \u003d √3/4 * a 2.
It is easily obtained from that which is known for an arbitrary figure with three sides. Just in the formula you need to take into account the fact that all sides of the triangle are equal.
More precisely, Heron's formula is required: S = √(p(p-a)(p-b)(p-c)). The value of the semi-perimeter for an equilateral triangle will be 3a/2. Thus, in each bracket under the root, the expression ((3a / 2) - a) will be obtained. It will give after transformation a/2.
Since there are three brackets, this expression will have a third degree. So, it will be transformed into a 3/8.
It still needs to be multiplied by the semi-perimeter, which is defined as the sum of the sides divided by 2. The expression will be: 3a 4 / 16. After extraction square root just the expression that is given in the first formula for the area of an equilateral triangle will remain.
Therefore, there is no need to memorize many formulas. You can just remember one - Heron. From it, by simple mathematical transformations, all the rest are obtained, for example, for an equilateral triangle.
Formula 2: given the radius of the inscribed circle
This expression is very similar to the previous entry. But still there are significant differences: a different letter is used, irrationality has gone into the denominator, a factor of 3 has appeared and the number 4 has disappeared. In general, it is easy to remember.
S = 3√3 * r2.
This formula is also easy to obtain from the one given for an arbitrary triangle. In it, the radius is multiplied by the sum of the sides and divided by 4. Since the sides have the same value, the sum will be replaced by 3a. Now we need to remove the "a" so that only the value of the radius remains. This will require an expression in which the side is divided by the product of 2 and the sine of the angle opposite the side. Since the angle is 60º, the value of the sine will be √3/2. Then the side is expressed in terms of the radius as follows: a = √3R. After a simple transformation, one can come to the expression for the area that was given at the beginning.
Formula 3: given the circumscribed circle and its radius
It is very similar to the first one. Only the number 3 appears in its numerator and the letter has changed to R.
S = 3√3/4 * R2.
Since the radius is twice as large as that considered in the previous paragraph, it is clear how it is obtained. It simply replaces r with R/2. And the necessary changes are being made.
Therefore, the formula can not be memorized. Just keep in mind the ratio of the radii of the inscribed and circumscribed circles around an equilateral triangle.
Formula 4: known height
In this case, the area of an equilateral triangle is:
S = n 2 / √3.
To understand how such a formula is obtained, you will again need to use the common one for all triangles. It looks like the product of the side by the height and by ½. Now, to find out the area of an equilateral triangle, you will have to remember or derive a mathematical expression for the height.
It is easy to find out if you use the fact that the height forms right triangle. This means that the height can be found as a leg - from the Pythagorean theorem. The second leg will be equal to half of the side, since the height is also the median (this is a well-known property of an equilateral triangle). Then the height will be determined as the square root of the difference of two squares. The first is "a", and the second is "a / 2". After raising to the second degree and extracting the root, it remains: n \u003d (√3 / 2) * a. From it a \u003d 2n / √3. After substituting it into the main formula for all triangles, you get the expression that is indicated at the beginning of the section.
Example #1
Condition. Calculate the area of an equilateral triangle if its side is known to be 4 cm.
Solution. Since the value of the sides of the figure is known, it is necessary to use the first formula.
First you need to square the number 4. From this action, the number 16 will be obtained. Now it is reduced with the four in the denominator. And as a result, 4 and √3 remain in the numerator, and the denominator becomes equal to one, which means that it can simply not be written down. This is the result that was required to be found in the problem.
Answer: 4√3 cm2.
Example #2
Condition. All sides of an equilateral triangle are 2√2 dm. Calculate its area.
Solution. The reasoning is the same as in the first problem. Only the value of the square of the side will be different. It needs to be separately erected in second degree 2 and irrationality. And the result will be: 4 * 2 = 8. After the reduction with the denominator, 2 and √3 remain in the numerator of the fraction, and the denominator disappears.
Answer: 2√3 dm 2 .
Example #3
Condition. A circle is inscribed in an equilateral triangle, its radius is 2.5 cm. It is necessary to calculate the area of the triangle.
Solution. To calculate the desired value, you will need to use the second formula.
First, the radius value must be squared. Get 6.25. Then this value must be multiplied by 3. The result of this action will be the number 18.75. But this is not the final value yet: it will contain the factor √3, which is present in the formula used.
Answer: 18.75√3 cm2.
Example #4
Condition. It is required to determine what is the area of an equilateral triangle, if its height is known - 3 dm.
Solution. Naturally, you need to choose the fourth formula. With its help, the easiest way to find the answer to this problem.
It is enough just to square the number 3, that is, the height, which will give the value 9. And then divide it by √3, which is in the formula.
Since it is not customary in mathematics to leave irrationality in the denominator of the answer, it must be eliminated. To do this, the fraction 9/√3 will need to be multiplied by a fraction with the same numerator and denominator, namely √3/√3. From this action, the value 9√3 will appear in the numerator, and the number 3 will appear in the denominator.
This fraction can and should be reduced by 3. This is the end result.
Answer: area - 3√3 dm 2.
Example #5
Condition. Given an equilateral triangle whose area is 27 cm 2. From this value, you need to find out the length of the side of the figure.
Solution. Since we are talking about a side, the first formula will do. From it, you can immediately derive a mathematical expression that will allow you to determine the side of the triangle.
To do this, the area must be multiplied by 4 and divided by the square root of three. This will give you the value for the side squared. To get just a side, you need to extract the root. The expression for the side will look like this: a = 2 * √(S/√3).
Since the area is known, you can immediately proceed to the calculations. The radical expression looks like a quotient of 27 and √3. We need to get rid of the irrationality in the denominator. You get 27√3 divided by 3. After the reduction, 1 remains in the denominator, which you can not write, and 9√3 remains in the numerator.
The next step is to extract the root from the resulting expression. The first factor gives a value of 3. But the second - √3 - requires attention. To make things easier, you can extract these roots and round the values.
√3 = 1.73; now we extract the root from it again and get 1.32.
It remains only to multiply it by 2 and get the desired result.
Answer: side is 2.64 cm.
