How to find the area of ​​a triangle ?

How to find the area of ​​a triangle ?

Mathematics - a complex science, requiring storage and the ability to operate a large number of formulas.Consider the specific situation, the task in front of you: find the area of ​​triangle ABC.Where to begin?

to any problem of this type of action is applicable scheme distinguish what is given (triangle type, data elements, etc.) - select the appropriate formula, which will allow for the original data to find the answer.So, we select the most common formula to answer the question, how to find the area of ​​a triangle:

  1. known at least one side of the triangle and the height drawn to it.In this case the classical formula helps S = ah / 2 .Here a - side length of the triangle, taken as the base, h - the height of the triangle length.It is important to select the height that is lowered to the ground.
  2. There are two sides of the triangle and the angle between them.Powered formula S = a * b * sin (β) / 2 .Here, a, b - the known length of the sides of a triangle, β - the angle between them.
  3. known all three sides of the triangle.Here, help formula Heron S = √ (p * (p-s1) * (p-s2) * (p-s3)) .Here, s1, s2, s3 - side of the triangle, p - semiperimeter.To find a half-perimeter, it is necessary to add up the lengths of all sides of the triangle and then divide in half.
  4. To find the area of ​​a right triangle, it is necessary to divide the work in half the lengths of the other two sides.This rule is used to solve problems on finding the area of ​​a triangle in the 4th grade school.If given a right triangle, then to calculate its area using the formula S = ab / 2 .Where a, b - catheti.
  5. To calculate the area of ​​an isosceles triangle is applicable formula of claim 1 - claim 3.Moreover, in the formula of claim 1 as a parameter h can serve besides height and median bisector becauseall elements are equal.
  6. If you know the coordinates of the vertices of the triangle on the plane, then use the formula
    S = | (Bx-Ax) (Cy-Ay) - (Cx-Ax) (By-Ay) | / 2 , where the vertices are given the coordinates of A (Ax, Ay), B (Bx, By), C (Cx, Cy).
  7. If the problem is given equilateral or right triangle with the known side a, the formula will help S = 2a * √3 / 4 .
  8. sided triangle area can be found using all formulas except claim 5, n7.

example.Find the area and its square for the right triangle with sides 2. We work according to claim 7: S = 2 * 2 * √3 / 4 = √3 (ed2).S2 = 3.

remains to note that in the above embodiments, the list does not end there.There are a lot of formulas for finding the area of ​​a triangle.Each task requires a careful analysis of the conditions, highlight the desired data for choosing the right solutions.Good luck in this search.