- values sides of the triangle
Some of the problems in geometry, or rather, by plane geometry, require finding the area of some given shape.The area can be any shape as the ultimate goal of the task, and intermediate calculations necessary to insert into a more complex formula.Often such problems are asked to find the area of a triangle.The initial values may be different.In some cases, known to any side of the triangle and the altitude drawn to it, in the other - the perimeter of the triangle, and so on.
Let's find the area of a triangle is given, if known to the three parties.To find the area of such a triangle is used the formula of Heron.To determine the area of this formula is needed to calculate the half-perimeter of the triangle first (n).Knowing the values of all three sides, to make it simple.It is necessary to sum up all the sides of the triangle - it will be his perimeter, and then divide the result by two.Thereafter, it is nec
received three of the difference should be multiplied together and the product is again multiplied by the value of the half-perimeter.After all of these steps and get the results of multiplication, it is necessary that the result of the square root.That number, which is obtained after extracting the square root, and is the area of a given triangle.If we write the brief, the formula area of a triangle is this: the area (S) = square root-tion of (n * (n-a) * (n-b) * (n-c)).As can be understood from the formula, the question of finding a triangle with the known values of the parties is very easy.
For example, how to find the area of a triangle if one knows the 3 sides: the side and is 3 centimeters, side b is equal to 4 centimeters, and a party with equal 2 centimeters.The perimeter of the triangle is equal to a + b + c = 3 cm + 4 cm + 2 cm = 9 cm So semiperimeter equals 9:. 2 = 4.5 santimetraPoluchim: S = square root of-tion (4.5 cm * (45 cm - 3 cm) * (4.5 cm - 4 cm) * (4.5 cm - 2 cm)) = 2.9 square centimeters
What if the value of the parties onlyIt is known, but also stated that they are equal in the problem statement?In such a case, how to find the area of a triangle if we know all the parties, and they are equal?You can, of course, also calculate it using the formula of Heron discussed above, but why the extra payments, if such a triangle is derived another formula, which is much simpler formula of Heron.According to this formula, you must first calculate the square root of-tion number 3, and then build a second degree of the length sides of the triangle, multiply the value in the second degree to the root of the number 3 and obtained by multiplying the product divided by the number 4. Get a predetermined area of a triangle.When recording, this formula is as follows: S = (a ^ 2 * root (3)) / 4
Let there be a triangle with equal sides of length equal to 3 centimeters.According to this formula can be obtained of the triangle area: S = (3 * 2 * root (3)) / 4 = 3.9 square centimeters.To check, correct or not calculated the value of a particular area of a triangle, it is possible to carry out additional calculations by formula Gerona and compare the results.
semiperimeter (n) = (3 + 3 + 3) / 2 = 4.5 cm.By Heron's formula is: S = square root of-tion (4.5 cm * (4.5 cm - 3 cm) * (4.5 cm - 3 cm) * (4.5 cm - 3 cm)) = 39 square centimeters.Both values of the area found by different formulas are identical.So the area of triangle is defined correctly.Solving some other tasks to consider in the data and use condition data corresponding to this formula.