Lesson Video: Finding the Area of a Triangle Using Trigonometry Mathematics

In this video, we will learn how to find the area of a triangle using the lengths of two sides and the sine of the included angle.

15:58

Video Transcript

In this video, we will learn how to find the area of a triangle using the lengths of two sides and the sine of the included angle. We have seen previously that we can find the area of a triangle using its base and perpendicular height. However, as these measurements are not always given, we will use our knowledge of the trigonometric ratios to derive the trigonometric formula for the area of a triangle.

Let’s begin by considering triangle 𝐴𝐡𝐢 as shown. We will let the lengths of the sides be lowercase π‘Ž, 𝑏, and 𝑐, where these side lengths are opposite their corresponding uppercase angles. If we know the lengths of two of the sides of the triangle together with their included angle, we will be able to derive the trigonometric formula. Let’s assume that we know the lengths of two sides π‘Ž and 𝑏 and the angle between them 𝐢.

Recalling that the usual formula for the area of a triangle is a half of its base multiplied by its perpendicular height, we can therefore draw a line from vertex 𝐡 that is perpendicular to the base 𝐴𝐢, and we will label this as β„Ž. Considering the right triangle 𝐡𝐷𝐢 together with the trigonometric ratio that states that sin πœƒ is equal to the opposite over the hypotenuse, we know that the sin of angle 𝐢 is equal to the opposite side β„Ž over the hypotenuse π‘Ž. Multiplying both sides of this equation by π‘Ž gives us β„Ž is equal to π‘Ž multiplied by sin 𝐢.

We can then substitute this expression for β„Ž into the general formula. The area of triangle 𝐴𝐡𝐢 is therefore equal to a half of 𝑏 multiplied by π‘Ž sin 𝐢, which can be rewritten as a half π‘Žπ‘ sin 𝐢. The trigonometric formula for the area of a triangle is a half π‘Žπ‘ sin 𝐢, where π‘Ž and 𝑏 are the lengths of two sides and 𝐢 is the measure of the included angle.

It is important to note that we can use any two sides together with the included angle. For example, in our diagram, a half 𝑏𝑐 sin 𝐴 and a half π‘Žπ‘ sin 𝐡 would also give us the area of the triangle. It is therefore better to not be overly concerned about the exact letters used and instead to understand what they represent in terms of the relative positioning of the sides and angle in the triangle.

We will now consider some specific examples.

𝐴𝐡𝐢 is a triangle, where 𝐡𝐢 equals 15 centimeters, 𝐴𝐢 equals 25 centimeters, and the measure of angle 𝐢 is 41 degrees. Find the area of 𝐴𝐡𝐢, giving your answer to three decimal places.

We will begin by sketching triangle 𝐴𝐡𝐢. We are told that the length of 𝐡𝐢 is 15 centimeters, 𝐴𝐢 is equal to 25 centimeters, and the measure of angle 𝐢 is 41 degrees. Since we are given the lengths of two sides of the triangle π‘Ž and 𝑏 together with their included angle 𝐢, we can calculate the area of the triangle using the formula a half π‘Žπ‘ sin 𝐢, where lowercase π‘Ž is equal to side 𝐡𝐢 and lowercase 𝑏 is equal to side 𝐴𝐢. The area of our triangle is therefore equal to a half multiplied by 15 multiplied by 25 multiplied by the sin of 41 degrees. Ensuring that our calculator is in degree mode, we can type this in directly, giving us 123.011067 and so on.

We are asked to give our answer to three decimal places. As the fourth number after the decimal point is a zero, we round down, giving us 123.011. Since the lengths of the triangles were given in centimeters, the area of triangle 𝐴𝐡𝐢 to three decimal places is 123.011 centimeters squared.

In this question, we were given two side lengths and their included angle. However, in our next example, we are given a slightly different set of information. This will require us to carry out some calculations before using the formula for the area of a triangle.

An isosceles triangle has two sides of length 48 centimeters and base angles of 73 degrees. Find the area of the triangle, giving the answer to three decimal places.

We begin by sketching the triangle, recalling that the base angles of an isosceles triangle are the angles formed by each of the equal sides with the third side. The two equal sides have length 48 centimeters, and the base angles of the triangle are 73 degrees. We are asked to find the area of the triangle. And one way to do this is using the formula a half π‘Žπ‘ sin 𝐢, where π‘Ž and 𝑏 are the lengths of two sides of the triangle and 𝐢 is the included angle between them.

If we label the triangle 𝐴𝐡𝐢 as shown, then since angles in a triangle sum to 180 degrees, the measure of angle 𝐢 is 180 degrees minus 73 degrees plus 73 degrees. This is the same as subtracting 146 degrees from 180 degrees. The measure of angle 𝐢 is therefore equal to 34 degrees. Substituting in the side lengths together with the included angle, we have the area of the triangle is equal to a half multiplied by 48 multiplied by 48 multiplied by the sin of 34 degrees. Typing this into our calculator in degree mode gives us 644.190224.

We are asked to round our answer to three decimal places. And since the fourth digit after the decimal point is a two, we round down. This gives us an answer of 644.190. The area of the isosceles triangle to three decimal places is 644.190 centimeters squared.

In our next example, we will work backwards to determine the length of a second side of a triangle when given the area, another side length, and the measure of one angle.

𝐴𝐡𝐢 is a triangle, where 𝐴𝐡 equals 18 centimeters, the measure of angle 𝐡 is 60 degrees, and the area of the triangle is 74 root three centimeters squared. Find length 𝐡𝐢, giving the answer to two decimal places.

We will begin by sketching the triangle 𝐴𝐡𝐢 using the information given. We are told that side length 𝐴𝐡 is equal to 18 centimeters. The measure of angle 𝐡 is 60 degrees. And the area of the triangle is 74 root three centimeters squared. We are asked to find the length of 𝐡𝐢.

We recall that we can calculate the area of any triangle when we are given the lengths of two sides together with the measure of the angle between them, known as the included angle. The formula for the area can be written a half π‘Žπ‘ sin 𝐡, where π‘Ž and 𝑐 are the two side lengths and 𝐡 is the included angle.

As already mentioned in this question, we know that the area of the triangle is 74 root three centimeters squared. This is therefore equal to a half multiplied by π‘₯ multiplied by 18 multiplied by sin of 60 degrees, where π‘₯ is the length of 𝐡𝐢 given in centimeters. 60 degrees is one of our special angles, and the sin of 60 degrees is root three over two. As a half of 18 is equal to nine, the right-hand side simplifies to nine root three over two multiplied by π‘₯.

In order to solve this equation for π‘₯, we can firstly divide through by root three. Multiplying both sides by two and dividing by nine gives us π‘₯ is equal to two multiplied by 74 over nine, or two-ninths of 74. This is equal to 148 over nine, or 16.4 recurring. Rounding to two decimal places, this is equal to 16.44. And we can therefore conclude that side length 𝐡𝐢 is equal to 16.44 centimeters to two decimal places.

In our final example, we will consider how we can use the trigonometric formula for the area of a triangle to calculate the area of a parallelogram.

𝐴𝐡𝐢𝐷 is a parallelogram, where 𝐴𝐡 is equal to 41 centimeters, 𝐡𝐢 is equal to 27 centimeters, and the measure of angle 𝐡 is 159 degrees. Find the area of 𝐴𝐡𝐢𝐷, giving the answer to the nearest square centimeter.

We will begin by sketching the parallelogram 𝐴𝐡𝐢𝐷 as shown. We are told that side 𝐴𝐡 is equal to 41 centimeters and side 𝐡𝐢 is equal to 27 centimeters. As the opposite sides are equal in length, we can add on the lengths of 𝐴𝐷 and 𝐷𝐢. We are also told that the measure of angle 𝐡 is 159 degrees.

We know that in order to calculate the area of any parallelogram, we can multiply the length of its base by its perpendicular height. However, in this question, we are not given the perpendicular height of the parallelogram. Instead, we notice that the parallelogram can be split into two congruent triangles: triangle 𝐴𝐡𝐢 and triangle 𝐴𝐷𝐢. As the triangles are congruent, the area of the parallelogram will be equal to twice the area of one of the triangles.

The area of any triangle can be calculated if we know the length of two sides together with the measure of the included angle such that the area is equal to a half π‘Žπ‘ sin 𝐡, where π‘Ž and 𝑐 are the lengths of two sides of the triangle and 𝐡 is the included angle between them. The area of triangle 𝐴𝐡𝐢 is therefore equal to a half multiplied by 27 multiplied by 41 multiplied by sin of 159 degrees. And since the area of the parallelogram is double this, it is equal to two multiplied by one-half multiplied by 27 multiplied by 41 multiplied by the sin of 159 degrees. After canceling a factor of two from the numerator and denominator, we can type this into our calculator, giving us 396.713 and so on.

We are asked to give our answer to the nearest square centimeter. And as the digit after the decimal point is greater than five, we round up. The area of the parallelogram 𝐴𝐡𝐢𝐷 to the nearest square centimeter is 397 square centimeters.

We will now finish this video by summarizing the key points. The area of any triangle can be calculated using the lengths of two of its sides and the sine of their included angle, such that the trigonometric formula for the area of a triangle is equal to a half π‘Žπ‘ sin 𝐢, where π‘Ž and 𝑏 are the lengths of two sides and 𝐢 is the measure of the included angle. We saw in this video that when given the area of a triangle and two pieces of information from the side lengths π‘Ž and 𝑏 and the angle 𝐢, the trigonometric formula can be used to find the missing side or angle measure. Finally, we saw that the trigonometric formula for the area of a triangle can be used to calculate the areas of other geometric shapes or compound shapes which can be divided into triangles.

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