12. Sine and the Area of a Triangle
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Chapter 2
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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Lesson Settings & Tools
| | 10 Theory slides |
| | 11 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Sine and the Area of a Triangle
Slide of 10
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.
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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.
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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 △ ABC, the lengths of AB and BC are 6 and 8 inches, respectively. The measure of ∠ ABC varies from 15^(∘) to 90^(∘).
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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 1 unit.
Izabella's homework problem is to find the area of this triangle. Help her find the answer.
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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 △ ABC needs to be found. Suppose line l bisects ∠ A and intersects AB at point M.
Since △ ABC is an equilateral triangle, the measure of each of its interior angles is 60 ^(∘). Consequently, ∠ ACM measures 30 ^(∘).
Considering △ ACM, it can be concluded that ∠ AMC is a right angle by the Triangle Angle Sum Theorem.
Consequently, CM is an altitude of △ ABC and AB its corresponding base. To find CM, the right triangle whose vertices are A, M, and C will be considered. Note that the hypotenuse of △ AMC is 1, and that CM is the opposite side to the angle whose measure is 60^(∘). Therefore, the sine ratio can be used to find CM. sin 60^(∘) = CM/1 ⇒ sin 60^(∘) = CM Recall that the exact value of sin 60^(∘) is sqrt(3)2. sin 60^(∘) = CM ⇒ CM=sqrt(3)/2 Now that the height was found, the area of △ ABC can be calculated.
The area of the equilateral triangle is sqrt(3)4 units^2.
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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 △ ABC, the lengths of AB and BC are 6 and 8 inches, respectively. The measure of ∠ ABC varies from 15^(∘) to 165^(∘).
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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.
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Hint
Begin by drawing the external altitude of the 40-foot side.
Solution
Begin by drawing the external altitude of the 40-foot side.
Notice that the angle opposite the altitude is the supplementary angle of 120^(∘). Its measure can be found by subtracting 120^(∘) from 180^(∘). 180^(∘)-120^(∘)=60^(∘) The triangle formed by the altitude, the side whose length is 30 feet, and the extension of the base, is a right triangle. Here, the hypotenuse is 30 feet and the measure of the opposite angle to the altitude is 60^(∘). Therefore, to find the height h, the sine ratio can be used. sin60^(∘)=h/30 The above equation can be solved for h.
sin60^(∘)=h/30
▼
Solve for h
MultEqn
LHS * 30=RHS* 30
30* sin60^(∘) = h
RearrangeEqn
Rearrange equation
h=30* sin60^(∘)
UseCalc
Use a calculator
h=2.598076...
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.
A=1/2bh
SubstituteII
b= 40, h= 2.598076...
A=1/2( 40)( 2.598076...)
▼
Evaluate right-hand side
MoveRightFacToNumOne
1/b* a = a/b
A=40/2(2.598076...)
CalcQuot
Calculate quotient
A=20(2.598076...)
UseCalc
Use a calculator
A=51.961524...
RoundInt
Round to nearest integer
A≈ 52
The area of the region is about 52 square feet.
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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.
Proof
To find the first formula, start by drawing the altitude from B and let h be its length. Since the altitude is perpendicular to the base, it generates two right triangles.
Because △ BCD is a right triangle, the height of the triangle can be related to the sine of ∠ C using the sine ratio. sin C = h/a ⇔ h = asin C Next, substitute the expression found for h into the general formula for the area of a triangle.
Area = 1/2bh ⇓ Area = 1/2absin C
The first formula was obtained. To obtain the second formula, notice that △ ABD is also a right triangle. Therefore, the sine ratio can be applied again, this time to connect h and ∠ A. sin A = h/c ⇔ h = csin A By substituting this expression into the general formula for the area of a triangle, the second formula can be obtained.
Area = 1/2bh ⇓ Area = 1/2bcsin A
To deduce the third formula, the altitude from C or A should be drawn. In this case, the altitude from C will be arbitrarily drawn and labeled D with a length of h. Because ∠ B is obtuse, the altitude will lie outside the triangle.
In this case, the length of the base is c and the height is h. Since △ BDC is a right triangle, the sine ratio can be used to connect ∠ CBD and h. sin CBD = h/a Since ∠ CBD and ∠ B 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 ∠ B and h can be written. sin B = sin CBD [0.5em] sin CBD = h/a ⇒ sin B = h/a Multiplying both sides of the last equation by a, it is obtained that h=asin B. Finally, substitute this expression for h into the formula for the area of △ ABC.
Area = 1/2ch ⇓ Area = 1/2acsin B
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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.
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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.
Area=1/2 a bsin C
Since the lengths of two sides and the measure of their included angle are known, a= 1110, b= 980, and C= 69.5 can be substituted into the formula to find the area.
Area=1/2absin C
SubstituteValues
Substitute values
Area=1/2( 1110)( 980)sin 69.5
▼
Evaluate right-hand side
Multiply
Multiply
Area=1/2(1 087 800)sin 69.5
MoveRightFacToNumOne
1/b* a = a/b
Area=1 087 800/2sin 69.5
CalcQuot
Calculate quotient
Area=543 900sin 69.5
UseCalc
Use a calculator
Area=509 456.003732...
RoundInt
Round to nearest integer
Area≈ 509 456
Therefore, the area of the Bermuda Triangle is approximately 509 456 square miles.
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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.
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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.
Write the answer correct to one decimal place.
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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.
To do so, the Triangle Angle Sum Theorem can be used. x^(∘)+17^(∘)+54^(∘) = 180^(∘) ⇓ x^(∘) = 109^(∘) 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.
A= 1/2(5)(4)sin109^(∘)
▼
Evaluate right-hand side
Multiply
Multiply
A= 1/2(20)sin109^(∘)
MoveRightFacToNumOne
1/b* a = a/b
A= 20/2sin109^(∘)
CalcQuot
Calculate quotient
A= 10sin109^(∘)
UseCalc
Use a calculator
A=9.455185...
RoundDec
Round to 1 decimal place(s)
A≈ 9.5
The area of the table is about 9.5 square feet. Therefore, Izabella needs about 9.5 square feet of fabric.
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Sine and the Area of a Triangle
Exercises
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Examining the diagram, we see that we know two sides and their included angle. Let's recall the formula of Area of a Triangle Using Sine. A=1/2absin C This formula allows for us to calculate the area of a triangle if we know the length of two sides a and b and the included angle C.
For this triangle, we also know two sides and one angle. Therefore, we can use the formula Area of a Triangle Using Sine, once again, to determine the area.
As in Part A and B, we have the necessary information to determine the triangle's area using sine.
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Solution
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In the given triangle, we know two angles. To be able to use the formula for calculating the triangle's area, we need to determine the remaining angle by using the Interior Angles Theorem. m∠ C + 38^(∘) +57^(∘) = 180^(∘) ⇓ m∠ C =85^(∘) Now that we know two sides and their included angle, we can determine the area of the triangle.
The area of the triangle is about 9.7 square centimeters.
Notice that we have been given the exterior angle of one of the angles in the triangle. Since these two angles form a linear pair, we can determine the inner angle by equating these angles with 180^(∘) and solving the resulting equation. m∠ C + 136^(∘) = 180^(∘) ⇓ m∠ C =44^(∘) Now we can calculate the area.
The triangle has an area of about 6.3 square centimeters.
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Solution
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By substituting the sides and the given area into the formula for calculating the Area of a Triangle Using Sine, we can solve for the included angle.
As we can see, the measure of the acute ∠ v is about 68^(∘).
We will solve for ∠ v in the same way as we did in Part A.
In this triangle, the measure of the acute angle labeled v is about 20^(∘).
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Solution
An equilateral triangle has sides with lengths of 6 centimeters. What is its area? Round the answer to one decimal place.
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An equilateral triangle is also equiangular. This means that it has three congruent angles, each with a measure of 60^(∘). Let's draw the triangle.
Let's determine the triangle's area by using the formula for the Area of a Triangle Using Sine.
The area of the triangle is about 15.6 square centimeters.
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Solution
What is the area of the quadrilateral in centimeters? Round the answer to one decimal place.
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To determine the area of the quadrilateral, we will divide it into two triangles and label them A_1 and A_2.
Notice that we now have two triangles where we know two sides and their included angle. This means we can calculate the area of the triangles using sine. Let's substitute each set of information into the formula for calculating the area. A_1&=1/2(4.2)(4.1)(sin 59^(∘)) [0.9em] A_2&=1/2(2.8)(2.2)(sin 108^(∘)) The sum of these equals the area of the quadrilateral, which we will label A_Q.
The quadrilateral's area is about 10.3 square centimeters.
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Solution
Sine and the Area of a Triangle
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