The area of the base of an equilateral triangle. What is and how to find the area of an equilateral triangle
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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 extracting the square root, 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 recognize if you use the fact that the height forms a 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 defined as Square root from 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. In it, you need to separately raise 2 and irrationality to the second power. 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 final 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. Insofar as we are talking about the side, then 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.
IN school course geometry, 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 learn, 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 a 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 right triangle, R - radius of the circumscribed circle, r - inscribed.
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 problem, including triangles, is related to 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 a circle and a 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.
An equilateral triangle is the simplest regular polygon of the possible. When finding its area, particular variants of its calculation arise. It is important to know and understand the signs and properties of this type of figure in order to easily calculate this parameter. All the methods presented below are quite simple to use and do not require deep thinking.
In contact with
Signs and properties of the figure
- Its value is the same in all cases and equals 60 degrees, regardless of the size of the sides.
- , height and median emitted from the same angle will match.
- Any side of an equilateral triangle equal to the other two.
- The center of a regular triangle will be the center for .
- Is a special case isosceles triangle.
Important! If at least one of these signs is observed, then the triangle is equilateral.
Equilateral triangle
Additionally this special case figure has the following properties:
Settlement through the side
There are many ways to calculate the area of this figure. All of them have their advantages and disadvantages. They are applied depending on the conditions presented to the task. The most popular way to find the desired value for an equilateral triangle is calculated through the product of half the sides and the sine of the angle between them, it looks like this: where, a and b are the sides, α is the angle between them.
In the case of equilateral, this method is simplified to a large extent. To do this, you need to refer to the features and properties discussed above. Based on the fact that all the angles of this figure are equal, and equal to 60 degrees. Sine 60 degrees, according to Bradis table, equals , transforming the original expression we get the following value: .
Considering that all sides of this figure are equal, the converted expression will give the following result: .
This formula is perfect if you know side size this figure. In this form, calculating this indicator is much easier and faster.
Those who remember Heron's formula know how to find the area of this figure. During the conversion, the expression will change to the one shown above. The area of this figure according to Heron is calculated as follows: , where, a, b, c are sides, and p is the semi-perimeter (). This expression is transformed quite simply. It is necessary to substitute the calculation of the semi-perimeter instead of the p value and gradually begin to reduce the expression. The sum of the sides can be represented as the sum of three identical sides and bring the reductions to the end. Mathematically it looks like this:
;
;
The resulting area formula and the functions below can only be used if the figure is correct otherwise it will not give the correct answer.
Calculating the area of a triangle given its side
Height calculation
You can also find the area of an equilateral triangle if you know its and side. Half the length of the height is multiplied by the side, any height and side can be selected, because according to the properties, they they are all the same: , where a is the length of the side. It is easy to remember, however, in practice it is used quite rarely.
If the task contains information that the triangle is equilateral and the height is known. And what is the length of the side is unknown, then you can use the formula that allows you to calculate it. Find a side can be divided by the double value of the height by the square root of three, mathematically it looks like this: . After that, the area formula is applied, where calculations are made through the side, it is described in the previous paragraph.
In order not to make unnecessary calculations, you can derive the formula for this indicator immediately through height. The square of the height is divisible by the square root of three. It will look like this: . In this case, you can not apply the formula of an isosceles triangle through the side.
Calculating the area of a triangle given its side and height
Calculation through circles
In mathematics, the method of calculating the value considered in the article by placing a figure in a circle or vice versa is also popular. Such a circle called described. If it is inside, then it is called inscribed. It is in this section that most questions arise, how to find the area of an equilateral polygon with three corners.
The circumscribed circle must pass across all peaks, the inscribed must pass through the sides only at one point along the tangent.
Drawing of an equilateral triangle circumscribed or inscribed in a circle
If the radius of the inscribed and circumscribed circles is given in the condition of the problem, then an expression can also be made from them, since together they will give the total length of the height. How the area is calculated using it is shown above: h \u003d R + r.
By transforming the formula, applying the calculation of the height h \u003d R + r, you can get the following value: . This formula can be simplified even more, because the radius of the circumscribed circle can be expressed through the radius of the inscribed. According to the properties of these circles, R = 2r, where r is the radius of the inscribed circle, R is the radius of the circumcircle. Respectively area of an equilateral triangle will be calculated like this:
If the size of the radius of the circumscribed circle is given, then the expression will look like this: .
Using these properties is useful for calculating the side of a figure. In order to find it, you can use the expression for the circumscribed circle, and for the inscribed one.
Given the radius of the circumscribed circle, you can find the desired value by raising the side to a cube, after which the result is divided by the radius increased 4 times. Mathematically, it can be written as follows: .
The process of calculating what the area of an equilateral triangle is equal to through any of the proposed formulas should not cause any particular difficulties. In order to successfully cope with this task, you do not need to memorize all of these methods, it is enough to remember the basic general calculation formulas, as well as the properties and features of this figure.
Attention! To check the correctness of the calculations, you can use several methods, the results must match.
Area of an equilateral triangle
Area of an equilateral triangle inscribed in a circle
Applying logical thinking, the calculations are easily converted into special cases, of which there are many more. It is inappropriate to fill your head with a lot of irrelevant information, it is better to develop a causal relationship for transforming expressions.
