In order to deal with the geometric and trigonometric problems you have to be aware of the formula for area of a triangle and know how to use it correctly. The formula for area of a triangle is the most common way to find the area of this kind of geometric figure. However, the method of calculating the area of triangle depends on what data, what numbers provides your geometric problem. The easiest formula to deal with the given task concerning the area of a triangle is to multiply the height and the base of the figure and divide the number that you got from multiplying by two. Apart from it, there are also other methods, taking into consideration what dimensions you are provided with. To choose the correct formula for area of a triangle you need to read the assignment carefully and determine what dimensions you already have and what you need to find out, whether you know the length of all the sides of the figure, the length of only one side, the angle, etc. In this article, you will learn the formula for area of a triangle and how to apply it to the common geometric problems, as well as get familiar with the essentials of geometry, trigonometry and some interesting facts about the history of triangles.
As we have already said, you need to use that or another formula for area of a triangle depending on the given in the task dimensions. The first and the easiest way to find the area of the triangle is to use the base and the height of the figure. In this case, you need to undertake the following steps:
- Find out the basis and the height of the given figure. By the base, one should mean the length of one side of the figure, that serves as the bottom line of the triangle. By the height, one should mean the length from the bottom side of the figure to the highest corner of the triangle. The height has to be perpendicular to the triangle. Keep in mind that the measure of the height may differ depending on whether you deal with the right triangle or with the non-right triangle.
- Write down the formula for area of a triangle that you are going to use. You will have to multiply the height and the base of the figure and divide the number that you got from multiplying by two. Once you have got the formula for area of a triangle written, you can proceed to the following step.
- Plug in the data that you already have concerning the base and the height. You need to identify these numbers and put them into the equation. For instance, if the height of the given triangle is 4 cm and the base of the figure is 6 cm, the formula for area of a triangle is going to look like this: area = ½ x (4cm x 6cm).
- Once you have your equation written, you can proceed to solving it. You can start with multiplying the height of the triangle and its base first of all and then divide the result by two, although whatever order of solving the equation you choose, the result will remain the same, as long as the order is correct.
Using the Heron's formula for area of a triangle
Another method to find the area of a triangle is to use the length of every side of the geometric figure. In this case, you will also have to apply the formula for area of a triangle, but it will differ from the formula for area of a triangle that you got familiar with earlier in the article. So, you will have to undertake the following steps:
- First of all, you will have to measure the semi-perimeter of the geometric figure. In order to calculate the semi-perimeter of a triangle, you need to get the sum of all the sides of the figure and divide the received result by two.
- Plug in the numbers indicating the lengths of all the sides of the triangle into the formula and calculate the semi-perimeter.
- Now that you have found the semi-perimeter of the triangle, you need to identify all the needed values for getting the area of the figure. Here, you are going to use the formula for area of a triangle that is known as Heron's formula.
- As for the order of the steps to undertake within using the formula for area of a triangle, you need to solve everything inside the parenthesis first of all, and then you will have to solve everything inside of the square root. The final step would be to deal with the square root itself.
Finding the area of an equilateral triangle
In case if the your assignment on geometry provides you with an equilateral triangle, all the previous methods for finding the area of the figure will not be really helpful. An equilateral triangle is a triangle, all sides of which are completely equal, as well as the angles. You will be aware of the fact that you are going to deal with this kind of triangle because of two reasons: you will get to know it from the information given in the assignment and because you know the fact that all the angles or all the sides of the given triangle are absolutely equal. In case if the assignment says that you have an equilateral triangle, and you are provided with the perimeter only, while the values of the sides of the figure are not given, all you need to do is to divide the perimeter by three. For instance, if the perimeter of an equilateral triangle is 15 cm, you need to divide 15 by 3 and you will find out that the length of one side of the triangle is five cm. Once you know the lengths of the sides of your triangle, you need to do the following:
- Write down the formula for area of a triangle. When dealing with this kind of geometric problem, the formula for area of a triangle will look like this: area = (s^2)(√3)/4. Here, the «s» will indicate the side of the geometric figure.
- The next step would be to plug in the value of the side length into the formula for area of a triangle that you are going to use. In case of this equation, the order of your operations will not influence the result, so you can proceed with whatever operation you want, remember being are attentive and careful.
- The received result should indicate the area of the given triangle. You can check the correctness of your result by doing the same operations in an opposite order. If you get the same numbers in the end, as you had in the beginning, it means that you did everything right.
Finding the area of a triangle using two sides and the included angle
The formula for area of a triangle when two sides and the included angle are used is not really the most popular, but it also works. First of all, you need to make sure that you know and understand the definition of the included angle. The term «including» indicates the angle allocated between the two sides of the geometric figure, the values of which are known. These values are very important to know because you will need to plug them into the formula for area of a triangle in order to get the required result. The values must be given in the assigned geometric problem, so read it carefully and find out all the needed data. Once you've got it written, you can proceed to undertaking the following steps:
- Write down the formula for area of a triangle. In case if you use the two sides of the triangle and the value of the included angle, the formula will represent the multiplying of the values of the two known sides of the figure and the sine of the know triangle and dividing the result by two. «B» and «c» will indicate the values of the sides and sinA will indicate the value of the indicated angle.
- The next step will be plugging the known values into the formula. For example, if the values of the sides are 25 and 55, the formula will look like this: area=1/2(25)(55) x sinA.
- Once you've got the values plugged into the formula, you have to solve the equation. First of all, you have to multiply the values of the known sides of the geometric figure and then, you will have to divide the result by two. After that, you need to multiply the received result by the sine of the indicated angle. As a matter of fact, while undertaking this step you may use the calculator instead of solving it on your own, which would save your time and efforts.
- As in all previously provided cases, you can check the correctness of the formula for area of a triangle and of the operations that you did by doing the same operations in the opposite order. If in the end you get the numbers equal to the numbers of the two sides and the indicated angle, the equation is solved correctly.
Interesting information about triangles
A triangle is a geometric figure that is characterized by have three edges and three vertices. A triangle is considered to be one of the basis and most important figures in geometry. There are different types of triangles, each of which depends on the properties and characteristics that or another triangle may have. All the triangles are divided into the groups according to the lengths of their sides, as well as according to the values of the internal angles of the figure.
In case if triangles are characterized according to the lengths of the sides of the figure, they are divided into the following categories:
- An equilateral triangle. This type of geometric figure can be distinguished due to the fact that all the sides of the figure have the same length. So, if you know the perimeter of such triangle and have to find the length of its sides, all you have to do is to divide the perimeter by three and the result will indicate the length of each side of the figure.
- An isosceles triangle. The specificity of this kind of a triangle is that it has two sides of the equal length. In addition, such triangle also has two angles of the same value.
- A scalene triangle. In this case, all the sides of the figure, as well as the angles, are of different value.
In case if triangles are characterized in accordance with their internal angles, the value of which is expressed in degrees, there are the following types of the figure:
- A right triangle. The specificity of this type of triangle is that one of his internal angles has to be of ninety degrees. An angle with the value of ninety degrees is called the «right» angle, and that is why this type of triangle also got the same name.
- In case if a triangle doesn't have any angle the value of which would be ninety degrees, this kind of triangle is called an oblique triangle.
- There are also such triangles with interior angles that have the value of less than ninety degrees. This type of geometry figure is called an acute triangle.
- A triangle which are characterized by having an angle of a value more than ninety degrees is called an obtuse triangle.
- In case if one angle of a triangle has the value of one hundred and eighty degrees, this type of geometric figure is called a degenerate triangle.
- There is also a type of triangle called a right degenerate triangle. Its specificity is that it has collinear vertices, the two of these vertices are coincident.
Apart from the fact that a triangle is an important shape in geometry, it also has a huge semantic history, which starts from the moment when Platon generated his geometrical theories. Apart from it, triangles played a significant role in the culture of Maya civilization and in India. A right triangle symbolizes beauty, strength, wisdom, and the rightness of everything (thinking, speaking, doing). Therefore, it has to be admitted that triangles are important not only in science, but also in religions of many cultures. Today the shape of a triangle is used for marketing due to its psychological affect on the perception of information and in many other areas.
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