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Similarity, Proof, and Trigonometry

Sine and the Area of a Triangle

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.
Multiply
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
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.
Evaluate right-hand side
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.
Evaluate right-hand side
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.

Explore

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