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
𝐴𝐵𝐶 as shown. We will let the lengths of the
sides be lowercase 𝑎, 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 𝑎 and 𝑏 and the angle between them 𝐶.
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 𝐵 that is perpendicular to the
base 𝐴𝐶, and we will label this as ℎ. Considering the right triangle
𝐵𝐷𝐶 together with the trigonometric ratio that states that sin 𝜃 is equal to the
opposite over the hypotenuse, we know that the sin of angle 𝐶 is equal to the
opposite side ℎ over the hypotenuse 𝑎. Multiplying both sides of this
equation by 𝑎 gives us ℎ is equal to 𝑎 multiplied by sin 𝐶.
We can then substitute this
expression for ℎ into the general formula. The area of triangle 𝐴𝐵𝐶 is
therefore equal to a half of 𝑏 multiplied by 𝑎 sin 𝐶, which can be rewritten as a
half 𝑎𝑏 sin 𝐶. The trigonometric formula for the
area of a triangle is a half 𝑎𝑏 sin 𝐶, where 𝑎 and 𝑏 are the lengths of two
sides and 𝐶 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 𝑎𝑐 sin 𝐵 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.
𝐴𝐵𝐶 is a triangle, where 𝐵𝐶
equals 15 centimeters, 𝐴𝐶 equals 25 centimeters, and the measure of angle 𝐶 is 41
degrees. Find the area of 𝐴𝐵𝐶, giving
your answer to three decimal places.
We will begin by sketching triangle
𝐴𝐵𝐶. We are told that the length of 𝐵𝐶
is 15 centimeters, 𝐴𝐶 is equal to 25 centimeters, and the measure of angle 𝐶 is
41 degrees. Since we are given the lengths of
two sides of the triangle 𝑎 and 𝑏 together with their included angle 𝐶, we can
calculate the area of the triangle using the formula a half 𝑎𝑏 sin 𝐶, where
lowercase 𝑎 is equal to side 𝐵𝐶 and lowercase 𝑏 is equal to side 𝐴𝐶. 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 𝐴𝐵𝐶 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 𝑎𝑏 sin 𝐶, where 𝑎 and 𝑏 are the lengths of two sides of the
triangle and 𝐶 is the included angle between them.
If we label the triangle 𝐴𝐵𝐶 as
shown, then since angles in a triangle sum to 180 degrees, the measure of angle 𝐶
is 180 degrees minus 73 degrees plus 73 degrees. This is the same as subtracting 146
degrees from 180 degrees. The measure of angle 𝐶 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.
𝐴𝐵𝐶 is a triangle, where 𝐴𝐵
equals 18 centimeters, the measure of angle 𝐵 is 60 degrees, and the area of the
triangle is 74 root three centimeters squared. Find length 𝐵𝐶, giving the answer
to two decimal places.
We will begin by sketching the
triangle 𝐴𝐵𝐶 using the information given. We are told that side length 𝐴𝐵
is equal to 18 centimeters. The measure of angle 𝐵 is 60
degrees. And the area of the triangle is 74
root three centimeters squared. We are asked to find the length of
𝐵𝐶.
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 𝑎𝑐 sin 𝐵, where 𝑎 and 𝑐 are the two side lengths and 𝐵 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 𝑥 multiplied by 18 multiplied by sin of 60 degrees, where 𝑥 is the
length of 𝐵𝐶 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 𝑥.
In order to solve this equation for
𝑥, we can firstly divide through by root three. Multiplying both sides by two and
dividing by nine gives us 𝑥 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 𝐵𝐶 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.
𝐴𝐵𝐶𝐷 is a parallelogram, where
𝐴𝐵 is equal to 41 centimeters, 𝐵𝐶 is equal to 27 centimeters, and the measure of
angle 𝐵 is 159 degrees. Find the area of 𝐴𝐵𝐶𝐷, giving
the answer to the nearest square centimeter.
We will begin by sketching the
parallelogram 𝐴𝐵𝐶𝐷 as shown. We are told that side 𝐴𝐵 is equal
to 41 centimeters and side 𝐵𝐶 is equal to 27 centimeters. As the opposite sides are equal in
length, we can add on the lengths of 𝐴𝐷 and 𝐷𝐶. We are also told that the measure
of angle 𝐵 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 𝐴𝐵𝐶 and
triangle 𝐴𝐷𝐶. 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 𝑎𝑐 sin 𝐵, where 𝑎 and 𝑐
are the lengths of two sides of the triangle and 𝐵 is the included angle between
them. The area of triangle 𝐴𝐵𝐶 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
𝐴𝐵𝐶𝐷 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 𝑎𝑏 sin 𝐶, where 𝑎 and 𝑏 are the lengths of two sides and 𝐶 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 𝑎
and 𝑏 and the angle 𝐶, 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.
