In order to cope with mathematical, particularly geometric and trigonometric problems, you need to know the triangle formula and understand how to use it appropriately. The triangle formula is considered to be the easiest method to find its area. At the same time, the methodology of calculating the triangle area can be different, taking into consideration the data provided in your assignment, especially the dimensions that you have. There is a number of different formulas that are used for coping with this deal, the easiest of which is to multiply the height and the base of the triangle and then, to divide the received number by two. In addition, there are also other ways, considering the data you possess and the specificities of the geometry assignment. To choose the correct triangle formula you need to read the geometry task attentively and figure out what dimensions you already have and what you need to calculate, whether the task provides all the sides of the triangle or only one side, whether it provides the angles or not and so on. In this article, you will get familiar with the triangle formula and learn how to use it when dealing with geometry assignments. Also, you will learn the basics of such branches of mathematics as geometry, trigonometry and get familiar with some fascinating facts concerning the history of triangles.
As it has already been mentioned, you have to apply that or another triangle formula concerning the dimensions provided in the assignment. The easiest and the most common method to find the triangle area is using such dimensions as the base and the height of the triangle. Therefore, if you choose to find the area by using the height and the base of the triangle, the best way for you will be to undertake the following steps:
- At first, you need to find the basis and the height of the triangle. The base is a term describing the length of the triangle, which is also its bottom line. The height of the given figure is its length from the bottom side right to the to the highest corner of the triangle. It is always perpendicular to the figure. Remember that the way of calculating the height can be different, which depends on the type of the triangle (whether it is right or non-right).
- Find out the triangle formula you are going to use and write it down. You will need to multiply the base of the figure and its height and divide the received number by two. This is how the triangle formula will look like. When you got it written, you can continue undertaking further steps.
- Write down the dimensions that you already have (the height of the triangle and its base). You have to determine this data and enter it into the equation. For example, if the triangle's base is 3 cm and the height of the given triangle is 5 cm, the triangle formula will be the following: area = ½ x (3cm x 5cm).
- Now that your equation is written, you may start solving the given problem. You may proceed with the following: multiply the base of the given figure and its height. After that, divide the received number by two. The order of the operations can be different, it doesn't matter.
Learning the Heron's triangle formula
There is also another way to find the triangle's area. Here, you will need to know the length of all its sides. This triangle formula differs from the previous formulas because it requires the usage of other dimensions. Read carefully the guidelines provided below:
- As for the start, you need to calculate the triangle's semi-perimeter. Here, you have to calculate the sum of all the sides first of all. And then, you need to divide the result that you will got by two.
- Enter the dimensions of the length of every triangle's side into the triangle formula. After that, find out the and semi-perimeter and write down the received number.
- Once you've got the semi-perimeter of the given figure, you have to determine the essential values in order to calculate the area of the triangle. This is where you will use the Heron's triangle formula.
- When using the triangle formula, keep in mind that there has to be specific order of operations. At first, you have to deal with the numbers within the parenthesis. After that, proceed to solving all what you have inside the square root. And only then, you will have to cope with the square root itself.
Dealing with an equilateral triangle
Sometimes you may have to deal with an equilateral triangle. In this case, everything that you have learnt before will not really help you a lot. When talking of an equilateral triangle, one should mean a geometry figure, the sides of which are all of the same dimensions, including the lengths of the sides and the degrees of the angles. There are two ways to find out, whether the triangle you have to deal with is equilateral or not. The first way is that you will be probably provided with the explanation of this issue right in the assignment. The second way is that if you know that all the angles of the triangle are equal, or that the lengths of all the sides are equal. The assignment may also provide you only with the perimeter of the figure, while saying that the triangle is equilateral. In this case there is still a simple way to find out the lengths of all the sides. Since they are all absolutely equal, you will only have to divide the given perimeter by three. For example, if the triangle's perimeter is 12 cm, all you have to do is to divide 12 by 3. Thus, you will know that the length of every side of the given triangle is 4 cm. Now that you've got all the needed dimensions, you can proceed to undertaking the following steps:
- Take a pen and a piece of paper and write down the triangle formula you are going to use.
- After that, enter the length of the side length into the triangle formula. You will get an equation, which you will need to solve.
- Once you've got your equation solved, the received result will show you the area of the triangle. If you want to make sure your operations were correct, just do the same operations once again but in the opposite order. If you will have the dimensions that were primarily provided in the assignment, everything is correct.
Another way to find the triangle's area
There is also another way to find the area of the given triangle, if none of the previously provided is appropriate for you. This method foresees the usage of two sides and the included angle between the given sides. This kind of triangle formula can be hardly called the most popular and commonly used, but it is also quite helpful. Before you use it, however, ensure that you know what the term «included angle» is. If before the word «angle» you see the term «included», it means that the given angle is located between the two sides of the triangle, the dimensions of which you are provided with. According to this triangle formula you need to use these dimensions in order to find out the area. If you are expected to use exactly this formula for the area, the needed dimensions must be provided in the assignment that you have to complete. Therefore, read the assignment very attentively and figure out the essential dimensions and write them down. After that, you will have to undertake the following steps:
- Jot down the triangle formula you are going to use. Since you are given the two sides of the figure and the included angle's value is also known, you need to do the next operation: multiply the sine of the included angle by the dimensions of the two given sides and then, divide the received number by two. Within your triangle formula, «b» and «c» will be the values of the triangle's sides and sinA will determine the value of the included angle.
- The second thing you need to do is entering the known dimensions into the triangle formula. For instance, if you got such values of the given sides as 15 cm and 20 cm, your formula should look like this: area=1/2(15)x sinA(20).
- Now that all the needed dimensions are plugged into the formula, your next step will be solving the equation. At first, the dimensions of the known sides should be multiplied by each other. Then, you need to divide the received result by two. Finally, you have to multiply the result by the sinA (the sin of the given indicated angle). When dealing with this step, you may turn to the help of calculator, which will make the task easier and save your time.
- When you've got the result that you have been looking for, you can check its rightness very easily. All you need to do is to repeat all the operations once more, but in the opposite order. If you get the same dimensions as those provided in the assignment, that you can be sure that you completed the task correctly.
What else you should know about triangles
Such geometry figure as triangle is recognized by the fact that it has three edges and three vertices. As a matter of fact, his figure is one of the basis geometry figures and it is used in geometry most often. While studying geometry, you will learn a lot of different types of triangles, which have various specificities and properties. All of these geometry figures can be classified in accordance with their characteristics, such as the length of the sides, of the internal angles and so on. In the first case, when triangles are classified according to the lengths of the sides, they are divided into the next groups:
- An equilateral triangle. This kind of a triangle foresees that all its sides have the same length, and the internal angles of this triangle are also the same. This is the easiest type of triangle to deal with, because knowing of only one dimension of it is enough to find out all the other dimensions.
- An isosceles triangle. You will know whether you deal with this type of triangle, if you know that it has two sides of equal length. Just like in the previous case, the angles of this triangle will be also the same.
- A scalene triangle. Here, you will never find two equal lengths of the sides or two equal values of the angles, because all the sides and angles in a scalene triangle are different.
If the classification of triangles is based according to the values of their internal angles, they will be divided into the following groups:
- A right triangle. In such triangle, one of the internal angles has to the value of 90 degrees. Since «right» is how we call such an angle, the triangle that has this angle is called quite the same.
- If there is no right angle in a triangle, it is called an oblique triangle. All the triangles without a right angle are called this way.
- If a triangle has angles the values of which are less than 90 degrees, such a triangle has to be called an acute triangle.
- If there is an angle in a triangle that has more than 90 degrees, this kind of triangle should be called an obtuse triangle.
- There are also triangles, where there is one angle with 180 degrees. If you deal with this kind of geometry figure, remember that it is called a degenerate triangle.
- Finally, you may face triangles with collinear vertices, where the two vertices are coincident. In this case, you deal with a right degenerate triangle.
Triangle has its long and interesting history not only as a geometry figure, but also as a great meaningful symbol, the importance of which is hard to overestimate in many world cultures and history of civilizations. People started researching and using triangles many thousands of years ago, they believed that a triangle is a symbol of wisdom, beauty and perfection (in case if the ides of the triangle are equal).
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