How to calculate the area of a triangle using trigonometry?

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How to calculate the area of a triangle using trigonometry?

Throughout geometry studies, which deal with various structures and shapes, you are required to calculate areas and perimeters. Each shape or structure has a different formula through which you can answer the question and calculate the area. Fortunately, there is one formula that can be applied to all triangles, and it can be used to calculate the area of a triangle using trigonometry.

In the field of mathematics, emphasis is also placed on trigonometry, which deals with the study of triangles, their angles, and sides. Every student is required to demonstrate knowledge of triangles (from right triangles to isosceles triangles), and thus also answer the question of how to calculate the area of a triangle using trigonometry.

One formula for all different triangles

Now that you know the formula for calculating the area of a triangle using trigonometry, you can use it in any question where you need to calculate areas in triangles. The formula for calculating the triangle:

Diagram of a triangle labeled ABC with sides AB = 5, AC = 8, and angle Y = 60°. The area formula  𝑆 𝐴𝐵𝐶 = (𝐴𝐵⋅𝐴𝐶⋅sin𝛾)/2 is shown.

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What is the area of the given triangle?

555999666

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How to calculate triangle area using trigonometry?

Throughout geometry studies, which deal with different structures and shapes, you are required to calculate areas and perimeters. Each shape or structure has a different formula through which you can answer the question and calculate the area. Fortunately, there is one formula that can be applied to all triangles. It can be used to calculate the area of a triangle using trigonometry.

In mathematics studies, emphasis is also placed on trigonometry, which deals with the study of triangles, their angles and sides. Both students studying in level B math in middle school, and those who take 3 units in high school, are required to demonstrate knowledge of triangles (from right triangles to isosceles triangles), and thus also answer the question of how to calculate the area of a triangle using trigonometry.

One formula for all different triangles

Now that you know the formula for calculating the area of a triangle using trigonometry, you can use it in any question where you need to calculate areas in triangles. The formula for calculating the triangle:

Diagram of a triangle labeled ABC with sides AB = 5, AC = 8, and angle Y = 60°. The area formula  𝑆 𝐴𝐵𝐶 = (𝐴𝐵⋅𝐴𝐶⋅sin𝛾)/2 is shown.

Example:

Given triangle ABCABC and it is known that:

Side ABAB equals 55

Side ACAC equals 88

Angle YY is 6060 degrees.

Let's insert the given values into the formula and we should obtain:

s=ACABsin602s =\frac {AC \cdot AB \cdot \sin60} {2}

In other words:

s=580.8662s =\frac {5\cdot 8\cdot 0.866} {2}

The result obtained is: 17.3217.32.

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Examples with solutions for Area of a Triangle

Exercise #1

What is the area of the given triangle?

555999666

Video Solution

Step-by-Step Solution

This question is a bit confusing. We need start by identifying which parts of the data are relevant to us.

Remember the formula for the area of a triangle:

A1- How to find the area of a triangleThe height is a straight line that comes out of an angle and forms a right angle with the opposite side.

In the drawing we have a height of 6.

It goes down to the opposite side whose length is 5.

And therefore, these are the data points that we will use.

We replace in the formula:

6×52=302=15 \frac{6\times5}{2}=\frac{30}{2}=15

Answer

15

Exercise #2

What is the area of the triangle in the drawing?

5557778.68.68.6

Video Solution

Step-by-Step Solution

First, we will identify the data points we need to be able to find the area of the triangle.

the formula for the area of the triangle: height*opposite side / 2

Since it is a right triangle, we know that the straight sides are actually also the heights between each other, that is, the side that measures 5 and the side that measures 7.

We multiply the legs and divide by 2

5×72=352=17.5 \frac{5\times7}{2}=\frac{35}{2}=17.5

Answer

17.5

Exercise #3

The triangle ABC is given below.
AC = 10 cm

AD = 3 cm

BC = 11.6 cm
What is the area of the triangle?

11.611.611.6101010333AAABBBCCCDDD

Video Solution

Step-by-Step Solution

The triangle we are looking at is the large triangle - ABC

The triangle is formed by three sides AB, BC, and CA.

Now let's remember what we need for the calculation of a triangular area:

(side x the height that descends from the side)/2

Therefore, the first thing we must find is a suitable height and side.

We are given the side AC, but there is no descending height, so it is not useful to us.

The side AB is not given,

And so we are left with the side BC, which is given.

From the side BC descends the height AD (the two form a 90-degree angle).

It can be argued that BC is also a height, but if we delve deeper it seems that CD can be a height in the triangle ADC,

and BD is a height in the triangle ADB (both are the sides of a right triangle, therefore they are the height and the side).

As we do not know if the triangle is isosceles or not, it is also not possible to know if CD=DB, or what their ratio is, and this theory fails.

Let's remember again the formula for triangular area and replace the data we have in the formula:

(side* the height that descends from the side)/2

Now we replace the existing data in this formula:

CB×AD2 \frac{CB\times AD}{2}

11.6×32 \frac{11.6\times3}{2}

34.82=17.4 \frac{34.8}{2}=17.4

Answer

17.4

Exercise #4

Complete the sentence:

To find the area of a right triangle, one must multiply ________________ by each other and divide by 2.

Step-by-Step Solution

To solve this problem, begin by identifying the elements involved in calculating the area of a right triangle. In a right triangle, the two sides that form the right angle are known as the legs. These legs act as the base and height of the triangle.

The formula for the area of a triangle is given by:

A=12×base×height A = \frac{1}{2} \times \text{base} \times \text{height}

In the case of a right triangle, the base and height are the two legs. Therefore, the process of finding the area involves multiplying the lengths of the two legs together and then dividing the product by 2.

Based on this analysis, the correct way to complete the sentence in the problem is:

To find the area of a right triangle, one must multiply the two legs by each other and divide by 2.

Answer

the two legs

Exercise #5

Calculate the area of the triangle below, if possible.

8.58.58.5777

Video Solution

Step-by-Step Solution

The formula to calculate the area of a triangle is:

(side * height corresponding to the side) / 2

Note that in the triangle provided to us, we have the length of the side but not the height.

That is, we do not have enough data to perform the calculation.

Answer

Cannot be calculated

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