How to calculate the area of a triangle using trigonometry?

🏆Practice area of a triangle

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.

Start practice

Test yourself on area of a triangle!

einstein

Complete the sentence:

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

Practice more now

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.

Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today
Test your knowledge

Examples with solutions for Area of a Triangle

Exercise #1

Calculate the area of the right triangle below:

101010666888AAACCCBBB

Video Solution

Step-by-Step Solution

Due to the fact that AB is perpendicular to BC and forms a 90-degree angle,

it can be argued that AB is the height of the triangle.

Hence we can calculate the area as follows:

AB×BC2=8×62=482=24 \frac{AB\times BC}{2}=\frac{8\times6}{2}=\frac{48}{2}=24

Answer

24 cm²

Exercise #2

Calculate the area of the triangle ABC using the data in the figure.

121212888999AAABBBCCCDDD

Video Solution

Step-by-Step Solution

First, let's remember the formula for the area of a triangle:

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

 

In the question, we have three pieces of data, but one of them is redundant!

We only have one height, the line that forms a 90-degree angle - AD,

The side to which the height descends is CB,

Therefore, we can use them in our calculation:

CB×AD2 \frac{CB\times AD}{2}

8×92=722=36 \frac{8\times9}{2}=\frac{72}{2}=36

Answer

36 cm²

Exercise #3

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 #4

Calculate the area of the following triangle:

444555AAABBBCCCEEE

Video Solution

Step-by-Step Solution

The formula for calculating the area of a triangle is:

(the side * the height from the side down to the base) /2

That is:

BC×AE2 \frac{BC\times AE}{2}

We insert the existing data as shown below:

4×52=202=10 \frac{4\times5}{2}=\frac{20}{2}=10

Answer

10

Exercise #5

Calculate the area of the following triangle:

666777AAABBBCCCEEE

Video Solution

Step-by-Step Solution

The formula for the area of a triangle is

A=hbase2 A = \frac{h\cdot base}{2}

Let's insert the available data into the formula:

(7*6)/2 =

42/2 =

21

Answer

21

Start practice
Related Subjects