If you have a need to find the area of a triangle, do not worry that you have long forgotten all the things that the teachers put into your head at school.Our article will tell you how to solve this question, and in different ways.
To begin with let us recall that the triangle is a figure which is formed at the intersection of three straight lines.The three points where the lines intersect, - the vertices of the figure, and the segments, opposing it, - a triangle edge.There are a number of particular types of triangles (isosceles, rectangular, equilateral), the area where we will also be looking for.
How to calculate the area of a triangle from the general formula
For the most general case of a given area of a geometric figure is calculated by the formula: Area = ½ length of one side of the figure, multiplied by the length of the height dropped to this side.
Find the area of a triangle when we know all three of his side
In that case, if you know all three sides of the triangle, the square, you can find it using the formula of Heron.To begin with we find semiperimeter triangle folded lengths of all three sides and dividing by two.Then, the square area is already, according to the following equation: SS = p (p-well) (b-p) (p), where a, b, c - the lengths of the sides figures, and p - the half-perimeter.To find the area of a square root of the resulting value.
Find the area of a triangle if we know its hypotenuse, a leg and formed their angle
For this we use the trigonometric sign and a formula:
S = 1/2 * a * b * sinB, where a and b - legwith the hypotenuse, and in - that the angle which is formed at their intersection.
In this formula, we can find the usual area of a triangle and an equilateral and isosceles and rectangular.
Find the area of a triangle, if we are aware of a leg and the angle opposite to it
formula is used: S = 1/2 (A * A) / (2tgB), where well - known leg, and B - corner him opposite.
find the area of a triangle, if only to know the hypotenuse and leg
first find the value FF = 1/2 (B * B - A * A).Then removed from the root of that number (F) and introduced into the formula for finding the area of a triangular shape: S = a * F.Here a - a leg in - hypotenuse.
find the area of a triangle if we know one of the sharp edges and the hypotenuse
Known conditional puzzles values into the formula: S = 1/2 (B * B) * cosA * sinA *.There is an acute angle - it's A, and - the hypotenuse.
Find the area of triangle vertex coordinates
If you conditional tasks given the coordinates of three points which are the vertices of a triangular shape, you can also calculate the area.
So, you are the top of the A (x1, y1) and B (x2, y2), B (x3, y3).To find the area use this formula: S = 1/2 ((x1-x3) (Y2-Y3) - (x2, x3) (V1-V3)).At the same time, remember that necessarily takes a module from the value that you calculate in brackets because some points may have coordinates with the sign "minus".
You can also operate in a different way.
Method 1. Find the length of the first all sides of the triangular shape, and then use Heron's formula, which has been described above.First, we find squares of the sides of such formulas:
AB * AB = (x1, x2) (x2-x1) + (Y1-Y2) (Y1-Y2);
BW * BW = (x2, x3) (x2, x3) + (v2, v3) (v2, v3);
VA * = VA (x3-x1) (x1-x3) + (V3-V1) (U3-U1).
find the half of the perimeter of the triangular shape:
p = 1 \ 2 (AB + BA + Ba)
Now substitute the values into the formula:
SS = p (p-AB) (p-BV) (p-VA).It turned out the area in the square.Extract from the value of the root and find, at last, what you were looking for.
way, out of curiosity, you can calculate the coordinates of the area of the above two methods.Then you will know that the totals will be slightly diverge.This happens because the result obtained in the first calculation will be rounded value, rather than the result obtained by the formula of Heron.Thus, to obtain more accurate data, it uses the second method.
