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12. Sine and the Area of a Triangle
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12. 

Sine and the Area of a Triangle

In geometry, the area of a triangle is traditionally calculated using its base and height. However, there's a trigonometric approach that offers a fresh perspective. By knowing the lengths of two sides of a triangle and the measure of their included angle, one can determine the triangle's area using the sine function. This method is especially useful when the height of the triangle is not directly provided or when it's challenging to determine. The sine-based formula provides a versatile tool for calculating areas, especially in complex geometric scenarios. For instance, considering real-world applications, this approach can be used to determine areas of irregular plots or regions, enhancing our understanding and application of geometry in practical situations.
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10 Theory slides
11 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Sine and the Area of a Triangle
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The area of a triangle is half the product of its base and the corresponding height. In this lesson, a formula for the area of a triangle involving the sine ratio will be derived.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Investigating the Area of a Triangular Table

Izabella has a triangular table in her kitchen. She wants to cover the surface of the table with a tablecloth. Two side lengths and two angle measures of the table are given.

Triangular Table
How much fabric does Izabella need to cover the surface of the table?
Explore

Investigating the Area of a Triangle

Typically to find the area of a triangle, the measurements of its height and base are used. On the other hand, if the lengths of two sides and the measure of their included angle are known, what should be the process to find the area of the triangle?

In the lengths of and are and inches, respectively. The measure of varies from to
What information is needed to find the area of How can the measure of be used to find the area of
Example

Area of an Equilateral Triangle

One way of finding the area of an equilateral triangle is by using trigonometric ratios.

Izabella wants to finish her math homework before she goes out to buy some fabric for her table. She is given an equilateral triangle whose side length is unit.

Equilateral Triangle

Izabella's homework problem is to find the area of this triangle. Help her find the answer.

Hint

What information is needed to find the area of a triangle?

Solution

The area of a triangle is half the product of its base and the corresponding height. Therefore, the height of needs to be found. Suppose line bisects and intersects at point

Since is an equilateral triangle, the measure of each of its interior angles is Consequently, measures

Considering it can be concluded that is a right angle by the Triangle Angle Sum Theorem.

Consequently, is an altitude of and its corresponding base. To find the right triangle whose vertices are and will be considered. Note that the hypotenuse of is and that is the opposite side to the angle whose measure is Therefore, the sine ratio can be used to find
Recall that the exact value of is
Now that the height was found, the area of can be calculated.
SubstituteII

,

▼
Multiply
IdPropMult

Identity Property of Multiplication

MultFrac

Multiply fractions

The area of the equilateral triangle is
Explore

Investigating the Area of Obtuse Triangles

In the previous example, it was shown that trigonometric ratios can be used to find the area of a triangle given the lengths of two sides and the measure of their included acute angle. What happens if the included angle is obtuse?

In the lengths of and are and inches, respectively. The measure of varies from to
How can the measure of be used to find the area of when is obtuse? What is the relation between the areas of when the measure of is degrees and degrees? What is the relation between the sine ratio of an acute angle and its supplementary angle?
Example

Finding the Area of an Obtuse Triangle

Izabella finished her homework and is ready to go. On her way out, she notices an empty triangular region in the garden.

Her father wants to fill this region with topsoil. To figure out how much topsoil he needs, he asks Izabella to find the area of the region. Help them find the area. If it is necessary, round the answer to the nearest square foot.

Hint

Begin by drawing the external altitude of the side.

Solution

Begin by drawing the external altitude of the side.

Notice that the angle opposite the altitude is the supplementary angle of Its measure can be found by subtracting from
The triangle formed by the altitude, the side whose length is feet, and the extension of the base, is a right triangle. Here, the hypotenuse is feet and the measure of the opposite angle to the altitude is Therefore, to find the height the sine ratio can be used.
The above equation can be solved for
▼
Solve for
MultEqn

RearrangeEqn

Rearrange equation

UseCalc

Use a calculator

Now that the height was found, the formula for the area of a triangle can be used to find the area of the triangular region.
SubstituteII

,

▼
Evaluate right-hand side
MoveRightFacToNumOne

CalcQuot

Calculate quotient

UseCalc

Use a calculator

RoundInt

Round to nearest integer

The area of the region is about square feet.
Discussion

Area of a Triangle Using Sine

There is another way of finding the area of a triangle that can be derived from the previous examples.

The area of a triangle is half the product of the lengths of any two sides and the sine of their included angle. There are three possible formulas for every triangle.
Triangle
The proof of this theorem will be developed using the triangle shown, where is obtuse. However, the same proof is valid for all triangles.

Proof

To find the first formula, start by drawing the altitude from and let be its length. Since the altitude is perpendicular to the base, it generates two right triangles.

Triangle with one altitude drawn
Because is a right triangle, the height of the triangle can be related to the sine of using the sine ratio.
Next, substitute the expression found for into the general formula for the area of a triangle.
The first formula was obtained. To obtain the second formula, notice that is also a right triangle. Therefore, the sine ratio can be applied again, this time to connect and
By substituting this expression into the general formula for the area of a triangle, the second formula can be obtained.

To deduce the third formula, the altitude from or should be drawn. In this case, the altitude from will be arbitrarily drawn and labeled with a length of Because is obtuse, the altitude will lie outside the triangle.

In this case, the length of the base is and the height is Since is a right triangle, the sine ratio can be used to connect and
Since and form a linear pair, they are supplementary angles. Recall that the sine of an angle is equal to the sine of its supplementary angle. With this information, and using the Substitution Property of Equality, a formula connecting and can be written.
Multiplying both sides of the last equation by it is obtained that Finally, substitute this expression for into the formula for the area of
Therefore, for any triangle, its area can be found by calculating half the product of the length of two of its sides and the sine of their included angle.
Example

Using Sine to Calculate Areas in the Real World

While sitting on the bus on the way to the fabric store, Izabella began daydreaming about a formula for the area of a triangle involving the sine ratio. She dreams about applying this formula to find the area of the Bermuda Triangle which she remembers from the movie Gulliver's Travels.

Given the lengths of two sides and the measure of their included angle, calculate the area of the Bermuda Triangle. Round the answer to the nearest square mile.

Hint

Use the formula for the area of a triangle involving the sine ratio.

Solution
Recall the formula for the area of a triangle involving the sine ratio.
Since the lengths of two sides and the measure of their included angle are known, and can be substituted into the formula to find the area.
SubstituteValues

Substitute values

▼
Evaluate right-hand side
Multiply

Multiply

MoveRightFacToNumOne

CalcQuot

Calculate quotient

UseCalc

Use a calculator

RoundInt

Round to nearest integer

Therefore, the area of the Bermuda Triangle is approximately square miles.
Pop Quiz

Practice Using Sine Finding the Area of a Triangle

In the following triangles, the lengths of two sides and the measure of their included angle is given. By using the formula derived earlier, find the area of the triangles. Round the answer to one decimal place.

Closure

Finding the Area of a Triangular Table

By using a formula derived in this lesson, the challenge presented at the beginning can be solved. Recall that the question asked what amount of fabric Izabella needs to cover the triangular table.

Triangular Table
Write the answer correct to one decimal place.

Hint

Begin by finding the included angle. Then use the formula for the area of a triangle using the sine ratio.

Solution

To find how much fabric Izabella needs, the surface area of the table must be found. Therefore, begin by finding the included angle.

Triangular Table
To do so, the Triangle Angle Sum Theorem can be used.
The lengths of two sides and the measure of their included angle are known. Therefore, the formula for the area of a triangle including the sine ratio can be used.
▼
Evaluate right-hand side
Multiply

Multiply

MoveRightFacToNumOne

CalcQuot

Calculate quotient

UseCalc

Use a calculator

RoundDec

Round to decimal place(s)

The area of the table is about square feet. Therefore, Izabella needs about square feet of fabric.



Level 1
Level 2
Level 3
Sine and the Area of a Triangle

Diego has bought a new trampoline where the bounce mat has the shape of a regular octagon.

a trampoline in the form of a circle with an octogon attached
The area of the octogon is square meter less than the area of the entire trampoline. Find the area of the octogon. Round the answer to the nearest square meter.

Let's label the radius of the trampoline r. This means the octagon's area, which we label A_O, can be expressed as the difference between the trampoline's area, which is a circle, and 1. A_O=π r^2-1 Since we do not know A_O or r, we need a second relation which contains these variables. Let's divide the octogon in 8 congruent isosceles triangles. These have a vertex angle of 360^(∘)8=45^(∘) and legs of r meters.

Now we can use the formula for calculating the area of a triangle using sine to determine an expression for the area of one these triangles. To determine the total area of the octagon A_O, we multiply this area by 8. A_O=8(1/2r(r)(sin 45^(∘))) ⇓ A_O=4sin 45^(∘) r^2 Now we have two equations which both contain A_O and r^2. By isolating r^2 in one of the equations, we can then substitute it into the other equation to determine A_O.

Now we can substitute r^2 for A_O+1π in the second equation and solve for A_O.

The octagon has an area of about 9 square meters.

Determine the length of the greater of the two unknown sides if the triangle's perimeter is centimeters and its area is square centimeters. Round the answer to the nearest centimeter.

a triangle with a side of 5 cm and an opposite angle of 108 degrees

Let's call the longer unknown side x and the shorter side we call y.

From the exercise, we know that the perimeter is 11 cm. Using this information we can write an equation. x+y+5=11 ⇔ x+y=6 We can also create a second equation containing x and y by using the fact that the area of the triangle is 3.8 square centimeters.

Now we have two equations which both describe the triangle in terms of x and y. This means we can create the following system of equations. x+y=6 & (I) 7.6=xysin 108^(∘) & (II) Let's solve for y in the first equation and substitute this into the second equation.

We can solve Equation (II) by using the Quadratic Formula.

To calculate x, we will use a graphing calculator to simplify the expression by calculating the value of the square root.

Now we can continue the quadratic formula.

As we can see, x could be either 4 centimeters or 2 centimeters depending on which side was labeled x. Since we labeled the longer side as x, we know that the longer side has a length of 4 centimeters.

What is the largest area the triangle in the diagram can have?

a triangle with two sides of 3 and 2 cm and an included angle of v degrees

Let's create an expression for the triangle's area.

The area is the product of 3 and sin v. Notice that 3 is a constant. Therefore, to find the greatest possible area, we have to maximize sin v. Recall that the sine value is measured on the vertical axis in the unit circle. Therefore, if we look at the unit circle, we notice that the greatest possible value is attained when v=90^(∘).

We see that the triangle's maximum area occurs when sin v=1 which happens when v=90^(∘).

The greatest possible area for the triangle is 3 square centimeters.

Sine and the Area of a Triangle

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