Find the area of an equilateral triangle by any formula for an arbitrary figure of this type or use those in which the feature of this particular figure has already been taken into account and the mathematical expressions are substantially simplified.
The first case only requires replacing all sides with the same value and taking into account that all the angles of the triangle are 60 °.Then it will remain to carry out simple transformations, which will lead to the formulas given in the finished form a little lower.
side is known. In this and the following formulas, the standard notation for the values of the triangle is adopted. More details can be found in the proposed table.
|radius of the circumferences of the inscribed and described||r and R, respectively|
Calculation of the area of the triangle in this case will be carried out using the form
S = √3 / 4 * a2.
It is easily obtained from the one that is known for an arbitrary three-sided figure. Just in the formula you need to take into account that all sides of the triangle are equal.
To be more precise, the Heron formula is required: S = √( p( p-a)( p-b)( p-c)).The semiperimeter value for an equilateral triangle is 3a / 2.Thus, in each parenthesis, the expression( (3a / 2) - a) is obtained under the root. It will give after the transformation a / 2.
Since there are three parentheses, this expression will have a third degree. So, it will be transformed into a3 / 8.
It still needs to be multiplied by a halfperimeter, which is defined as the sum of the sides divided by 2. The result is 3a4 / 16.After extracting the square root, exactly the expression that is given in the first formula for the area of an equilateral triangle remains.
Therefore, there is no need to memorize many formulas. You can just remember one - Heron. From it, by simple mathematical transformations, all the others 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 record. But still there are significant differences: another letter is used, irrationality has gone to the denominator, a multiplier 3 has appeared and the figure has disappeared 4. In general, it is easy to remember.
S = 3√3 * r2.
This formula can also be easily obtained from the one given for an arbitrary triangle. In it, the radius is multiplied by the sum of the sides and is divided by 4. Since the sides have the same value, the sum is replaced by 3a. Now we need to remove "a", so that only the radius value remains. This requires an expression in which the side is divided by the product 2 and the sine of the opposite side of the angle. Since the angle is 60º, the sine value will be √3 / 2.Then the side is expressed in terms of the radius as follows: a = √3R.After a simple transformation, one can arrive at that expression for the area, which is given at the beginning.
Formula 3: given circumscribed circle and its radius
It is very similar to the first one. Only in its numerator appears the number 3 and the letter is changed to R.
S = 3√3 / 4 * R2.
Since the radius is twice that of the one considered in the previous paragraph, it is clear how it works. It simply puts R / 2 instead of r. And necessary transformations are carried out.
Therefore, the formula can not be memorized. Just keep in mind the ratio of the radii of the inscribed and described around an equilateral triangle of circles.
Formula 4: known height
In this case, the area of an equilateral triangle is:
S = n2 / √3.
To understand how this formula turns out, we will again need to use the common for all triangles. It looks like the work of the party to the height and to ½.Now, in order to know the area of an equilateral triangle, one has to recall or derive a mathematical expression for height.
It's easy to find out if you take advantage of the fact that the height forms a right-angled triangle. Hence, the height can be found as a cathet - from the theorem of Pythagoras. The second leg is equal to half the side, since the height is also a median( this is a known property of an equilateral triangle).Then the height will be defined as the square root of the difference of two squares. The first "a", and the second "a / 2".After raising to the second power and extracting the root, it remains: n =( √3 / 2) * a. From it, a = 2n / √3.After substituting it into the formula basic for all triangles, we get the expression that is indicated at the beginning of the section.
Example No. 1
Condition. Calculate the area of an equilateral triangle if it is known that its side has a value of 4 cm.
Solution. Since the value of the sides of the figure is known, it is necessary to use the first formula.
First we need to square the number 4. This action will result in the number 16. Now it is shortened with the four in the denominator. As a result, the numerator remains 4 and √3, and the denominator becomes equal to one, which means that it can simply not be recorded. This is the result, which was to be found in the problem.
Answer: 4√3 cm2.
Example No. 2
Condition. All sides of an equilateral triangle are equal to 2√2 dm. Calculate its area.
Solution. The reasoning is the same as in the first problem. Only the value of the side square will be different. In it it is necessary to raise separately to the second power 2 and irrationality. And the result will be: 4 * 2 = 8. After the abbreviation with the denominator, 2 and √3 remain in the numerator of the fraction, and the denominator disappears.
Answer: 2√3 dm2.
Example No. 3
Condition. An equilateral triangle is inscribed with a circle, its radius is 2.5 cm. It is necessary to calculate the area of the triangle.
Solution. To calculate the required value, you need to use the second formula.
First, the radius value must be squared. It will turn out to be 6.25.Then this value needs to be multiplied by 3. The result of this action will be the number 18.75.But this is not a final value: it will have a factor of √3, which is present in the formula used.
Answer: 18,75√3 cm2.
Example No. 4
Condition. It is required to determine what the area of an equilateral triangle is equal to, if its height is known - 3 dm.
Solution. Naturally, you need to choose the fourth formula. With its help, the easiest way is to find the answer to this problem.
It is enough just to square the number 3, that is, the height that 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, then it is necessary to get rid of it. For this, the fraction 9 / √3 must be multiplied by a fraction with the same numerator and denominator, namely √3 / √3.From this action, the number 9.3 appears in the numerator, and the number will appear in the denominator 3.
This fraction can and should be cut by 3. This is the final result.
Answer: area - 3√3 dm2.
Example No. 5
Condition. An equilateral triangle with an area of 27 cm2 is given. By this value you need to know the length of the side of the figure.
Solution. As far as the side is concerned, the first formula is suitable. From it you can immediately derive a mathematical expression that will allow you to determine the side of the triangle.
For this, the area should be multiplied by 4 and divided by the square root of the three. This will be the value for the side in the square. To get a simple 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 begin to calculate. The root expression looks like a quotient 27 and √3.It is necessary to get rid of the irrationality in the denominator. The result is 27√3, divided by 3. After the abbreviation, 1 remains in the denominator, which can not be written, but the numerator remains 9√3.
The next step is to extract the root from the resulting expression. The first factor gives the value 3. But the second - √3 - requires attention. To simplify the task, you can extract these roots and round off the values.
√3 = 1.73;Now we extract the root from it and get 1.32.
It only remains to multiply it by 2 and get the desired result.
Answer: side is equal to 2.64 cm.