How to find the area of a triangle?
Mathematics is a complex science that requires memorization and the ability to operate with a large number of formulas. Consider a specific situation, before you task: find the area of the triangle ABC. Where to begin?
The action scheme is applicable to any task of this type: select what is given (triangle type, given elements, etc.) - choose the appropriate formula that will allow you to find the answer from the source data. So, we single out the most common formulas for answering the question of how to find the area of a triangle:
- At least one side of the triangle and the height drawn to it are known. In this case, the classic formula will help.S = ah / 2. Here a is the length of the side of the triangle, taken as the base, h is the length of the height of the triangle. It is important to choose the height that is lowered exactly to the base.
- There are two sides of the triangle and the angle between them. Formula worksS = a * b * sin (β) / 2. Here a, b are the known lengths of the sides of the triangle, β is the angle between them.
- All three sides of the triangle are known. Here will help the formula GeronaS = √ (p * (p-s1) * (p-s2) * (p-s3)). Here s1, s2, s3 are sides of a triangle, p is a semi-perimeter. To find a semi-perimeter, you need to add the lengths of all sides of the triangle and divide the sum in half.
- To find the area of a right-angled triangle, it is necessary to divide the product of the lengths of its legs in half. Such a rule is used to solve problems of finding the area of a triangle already in the 4th grade of a school. If a right triangle is given, then to calculate its area we use the formulaS = ab / 2. Here a, b are the legs.
- To calculate the area of an isosceles triangle, the applicable formula is paragraph 1 - paragraph 3. Moreover, in the formula of claim 1, in addition to height and the median, the bisector, can act as a parameter h, since all elements are equal.
- If the coordinates of the vertices of the triangle on the plane are known, then we use the formula
S = | (Bx-Ax) (Cy-Ay) - (Cx-Ax) (By-Ay) | / 2where the vertices are given by the coordinates A (Ax, Ay), B (Bx, By), C (Cx, Cy).
- If the problem is given an equilateral or regular triangle with a known side a, the formula will helpS = 2a * √3 / 4.
- The area of a multi-sided triangle can be found using all formulas, except for item 5, item 7.
Example. Find the area and its square for a regular triangle with side 2. We work according to claim 7: S = 2 * 2 * √3 / 4 = √3 (units2). S2=3.
It remains to be noted that the list does not end with the listed options. There are many formulas for finding the area of a triangle. Each task requires a careful analysis of the conditions, the allocation of the necessary data to select the correct solution. Best of luck with this search.
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