NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area
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Lesson 31: Using Trigonometry to Determine Area
Student Outcomes
Students prove that the area of a triangle is one-half times the product of two side lengths times the sine of
the included angle and solve problems using this formula.
Students find the area of an isosceles triangle given the base length and the measure of one angle.
Lesson Notes
Students discover how trigonometric ratios can help with area calculations in cases where the measurement of the
height is not provided. In order to determine the height in these cases, students must draw an altitude to create right
triangles within the larger triangle. With the creation of the right triangles, students can then set up the necessary
trigonometric ratios to express the height of the triangle (G-SRT.C.8). Students carefully connect the meanings of
formulas to the diagrams they represent (MP.2 and 7). In addition, this lesson introduces the formula Area =12
๐๐ sin ๐ถ
as described by G-SRT.D.9.
Classwork
Opening Exercise (5 minutes)
Opening Exercise
Three triangles are presented below. Determine the areas for each triangle, if possible. If it is not possible to find the
area with the provided information, describe what is needed in order to determine the area.
The area of โณ ๐จ๐ฉ๐ช is ๐
๐(๐)(๐๐), or ๐๐ square units, and the area of โณ ๐ซ๐ฌ๐ญ is
๐
๐(๐)(๐๐), or ๐๐ square units. There is
not enough information to find the height of โณ ๐ฎ๐ฏ๐ฐ and, therefore, the area of the triangle.
Is there a way to find the missing information?
Without further information, there is no way to calculate the area.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area
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Example 1 (13 minutes)
What if the third side length of the triangle were provided? Is it possible to determine the area of the triangle
now?
Example 1
Find the area of โณ ๐ฎ๐ฏ๐ฐ.
Allow students the opportunity and the time to determine what they must find (the height) and how to locate it (one
option is to drop an altitude from vertex ๐ป to side ๐บ๐ผฬ ฬ ฬ ). For students who are struggling, consider showing just the
altitude and allowing them to label the newly divided segment lengths and the height.
How can the height be calculated?
By applying the Pythagorean theorem to both of the created right triangles to find ๐ฅ,
โ2 = 49 โ ๐ฅ2 โ2 = 144 โ (15 โ ๐ฅ)2
49 โ ๐ฅ2 = 144 โ (15 โ ๐ฅ)2
49 โ ๐ฅ2 = 144 โ 225 + 30๐ฅ โ ๐ฅ2
130 = 30๐ฅ
๐ฅ =13
3
๐ป๐ฝ =13
3, ๐ผ๐ฝ =
32
3
The value of ๐ฅ can then be substituted into either of the expressions equivalent to โ2 to find โ.
โ2 = 49 โ (13
3)
2
โ2 = 49 โ169
9
โ =4โ17
3
MP.13
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area
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What is the area of the triangle?
Area = (1
2) (15) (
4โ17
3)
Area = 10โ17
Discussion (10 minutes)
Now, consider โณ ๐ด๐ต๐ถ, which is set up similarly to the triangle in Example 1:
Write an equation that describes the area of this triangle.
Area =12
๐โ
Write the left-hand column on the board, and elicit the anticipated student response on the right-hand side after writing
the second line; then, elicit the anticipated student response after writing the third line.
We will rewrite this equation. Describe what you see.
Area =1
2๐โ
Area =1
2๐โ (
๐
๐)
Area =1
2๐๐ (
โ
๐)
The statement is multiplied by 1.
The last statement is rearranged, but the value remains the
same.
Create a discussion around the third line of the newly written formula. Modify โณ ๐ด๐ต๐ถ: Add in an arc mark at vertex ๐ถ,
and label it ๐.
What do you notice about the structure of โ
๐? Can we think of this newly written area formula in a different
way using trigonometry?
The value of โ
๐ is equivalent to sin ๐; the
newly written formula can be written as
Area =12
๐๐ sin ๐.
If the area can be expressed as 1
2๐๐ sin ๐, which
part of the expression represents the height?
โ = ๐ sin ๐
Scaffolding:
For students who are ready for a challenge, pose the unstructured question, โHow could you write a formula for area that uses trigonometry, if we didnโt know โ?โ
For students who are struggling, use a numerical example, such as the 6โ8โ10 triangle in Lesson 24, to demonstrate how โ can be found using sin ๐.
MP.2 &
MP.7
๐
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area
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Compare the information provided to find the area of โณ ๐บ๐ป๐ผ in the Opening Exercise, how the information
changed in Example 1, and then changed again in the triangle above, โณ ABC.
Only two side lengths were provided in the Opening Exercise, and the area could not be found because
there was not enough information to determine the height. In Example 1, all three side lengths were
provided, and then the area could be determined because the height could be determined by applying
the Pythagorean theorem in two layers. In the most recent figure, we only needed two sides and the
included angle to find the area.
If you had to determine the area of a triangle and were given the option to have three side lengths of a triangle
or two side lengths and the included measure, which set of measurements would you rather have? Why?
The response to this question is a matter of opinion. Considering the amount of work needed to find the area when
provided three side lengths, students should opt for the briefer, trigonometric solution!
Example 2 (5 minutes)
Example 2
A farmer is planning how to divide his land for planting next yearโs crops. A triangular plot of land is left with two known
side lengths measuring ๐๐๐ ๐ฆ and ๐, ๐๐๐ ๐ฆ.
What could the farmer do next in order to find the area of the plot?
With just two side lengths known of the plot of land, what are the farmerโs options to determine the area of
his plot of land?
He can either measure the third side length, apply the Pythagorean theorem to find the height of the
triangle, and then calculate the area, or he can find the measure of the included angle between the
known side lengths and use trigonometry to express the height of the triangle and then determine the
area of the triangle.
Suppose the included angle measure between the known side lengths is 30ยฐ. What is the area of the plot of
land? Sketch a diagram of the plot of land.
Area =12
(1700)(500) sin 30
Area = 212โ500
The area of the plot of land is
212,500 square meters.
Exercise 1 (5 minutes)
Exercise 1
A real estate developer and her surveyor are searching for their next piece of land to build on. They each examine a plot
of land in the shape of โณ ๐จ๐ฉ๐ช. The real estate developer measures the length of ๐จ๐ฉฬ ฬ ฬ ฬ and ๐จ๐ชฬ ฬ ฬ ฬ and finds them to both be
approximately ๐, ๐๐๐ feet, and the included angle has a measure of approximately ๐๐ยฐ. The surveyor measures the
length of ๐จ๐ชฬ ฬ ฬ ฬ and ๐ฉ๐ชฬ ฬ ฬ ฬ and finds the lengths to be approximately ๐, ๐๐๐ feet and ๐, ๐๐๐ feet, respectively, and measures
the angle between the two sides to be approximately ๐๐ยฐ.
MP.2 &
MP.7
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area
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a. Draw a diagram that models the situation, labeling all lengths and angle measures.
b. The real estate developer and surveyor each calculate the area of the plot of land and both find roughly the
same area. Show how each person calculated the area; round to the nearest hundred. Redraw the diagram
with only the relevant labels for both the real estate agent and surveyor.
๐จ =๐
๐(๐๐๐๐)(๐๐๐๐) ๐ฌ๐ข๐ง ๐๐
๐จ โ ๐โ๐๐๐โ๐๐๐
The area is approximately ๐, ๐๐๐, ๐๐๐ square feet.
๐จ =๐
๐(๐๐๐๐)(๐๐๐๐) ๐ฌ๐ข๐ง ๐๐
๐จ โ ๐โ๐๐๐โ๐๐๐
The area is approximately ๐, ๐๐๐, ๐๐๐ square feet.
c. What could possibly explain the difference between the real estate agentโs and surveyorโs calculated areas?
The difference in the area of measurements can be accounted for by the approximations of the measurements
taken instead of exact measurements.
Closing (2 minutes)
Ask students to summarize the key points of the lesson. Additionally, consider asking students to respond to the
following questions independently in writing, to a partner, or to the whole class.
For a triangle with side lengths ๐ and ๐ and included angle of measure ๐, when will we need to use the area
formula Area =12
๐๐ sin ๐ ?
We will need it when we are determining the area of a triangle and are not provided a height.
Recall how to transform the equation Area =12
๐โ to Area =12
๐๐ sin ๐.
Exit Ticket (5 minutes)
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area
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Name Date
Lesson 31: Using Trigonometry to Determine Area
Exit Ticket
1. Given two sides of the triangle shown, having lengths of 3 and 7 and their included angle of 49ยฐ, find the area of the
triangle to the nearest tenth.
2. In isosceles triangle ๐๐๐ , the base ๐๐ = 11, and the base angles have measures of 71.45ยฐ. Find the area of โณ ๐๐๐
to the nearest tenth.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area
472
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Exit Ticket Sample Solutions
1. Given two sides of the triangle shown, having lengths of ๐ and ๐ and their included angle of ๐๐ยฐ, find the area of the
triangle to the nearest tenth.
๐๐ซ๐๐ =๐
๐(๐)(๐)(๐ฌ๐ข๐ง ๐๐)
๐๐ซ๐๐ = ๐๐. ๐(๐ฌ๐ข๐ง ๐๐) โ ๐. ๐
The area of the triangle is approximately ๐. ๐ square units.
1. In isosceles triangle ๐ท๐ธ๐น, the base ๐ธ๐น = ๐๐, and the base angles have measures of ๐๐. ๐๐ยฐ. Find the area of
โณ ๐ท๐ธ๐น to the nearest tenth.
Drawing an altitude from ๐ท to midpoint ๐ด on ๐ธ๐นฬ ฬ ฬ ฬ cuts
the isosceles triangle into two right triangles with
๐ธ๐ด = ๐ด๐น = ๐. ๐. Using tangent, solve the following:
๐ญ๐๐ง ๐๐. ๐๐ =๐ท๐ด
๐. ๐
๐ท๐ด = ๐. ๐(๐ญ๐๐ง ๐๐. ๐๐)
๐๐ซ๐๐ =๐
๐๐๐
๐๐ซ๐๐ =๐
๐(๐๐)(๐. ๐(๐ญ๐๐ง ๐๐. ๐๐))
๐๐ซ๐๐ = ๐๐. ๐๐(๐ญ๐๐ง ๐๐. ๐๐) โ ๐๐. ๐
The area of the isosceles triangle is approximately ๐๐. ๐ square units.
Problem Set Sample Solutions
Find the area of each triangle. Round each answer to the nearest tenth.
1.
๐๐ซ๐๐ =๐
๐(๐๐)(๐) (๐ฌ๐ข๐ง ๐๐)
๐๐ซ๐๐ = ๐๐(๐ฌ๐ข๐ง ๐๐) โ ๐๐. ๐
The area of the triangle is approximately ๐๐. ๐ square
units.
2.
๐๐ซ๐๐ =๐
๐(๐)(๐๐)(๐ฌ๐ข๐ง ๐๐)
๐๐ซ๐๐ = ๐๐(๐ฌ๐ข๐ง ๐๐) โ ๐. ๐
The area of the triangle is approximately ๐. ๐ square units.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area
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3.
๐๐ซ๐๐ =๐
๐(๐) (๐
๐
๐) (๐ฌ๐ข๐ง ๐๐)
๐๐ซ๐๐ = ๐๐(๐ฌ๐ข๐ง ๐๐) โ ๐๐. ๐
The area of the triangle is approximately ๐๐. ๐ square units.
4.
The included angle is ๐๐ยฐ by the angle sum of a triangle.
๐๐ซ๐๐ =๐
๐(๐๐)(๐ + ๐โ๐) ๐ฌ๐ข๐ง ๐๐
๐๐ซ๐๐ = ๐(๐ + ๐โ๐) (โ๐
๐)
๐๐ซ๐๐ = (๐๐ + ๐๐โ๐) (โ๐
๐)
๐๐ซ๐๐ = ๐๐โ๐ + ๐๐(๐)
๐๐ซ๐๐ = ๐๐โ๐ + ๐๐ โ ๐๐. ๐
The area of the triangle is approximately ๐๐. ๐ square units.
5. In โณ ๐ซ๐ฌ๐ญ, ๐ฌ๐ญ = ๐๐, ๐ซ๐ญ = ๐๐, and ๐โ ๐ญ = ๐๐ยฐ. Determine the area of the triangle. Round to the nearest tenth.
๐๐ซ๐๐ =๐๐
(๐๐)(๐๐)๐ฌ๐ข๐ง(๐๐) โ ๐๐๐. ๐
The area of โณ ๐ซ๐ฌ๐ญ is ๐๐๐. ๐ units2.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area
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6. A landscape designer is designing a flower garden for a triangular area that is bounded on two sides by the clientโs
house and driveway. The length of the edges of the garden along the house and driveway are ๐๐ ๐๐ญ. and ๐ ๐๐ญ.,
respectively, and the edges come together at an angle of ๐๐ยฐ. Draw a diagram, and then find the area of the garden
to the nearest square foot.
The garden is in the shape of a triangle in which the lengths of two sides and the included angle have been provided.
๐๐ซ๐๐(๐จ๐ฉ๐ช) =๐
๐(๐ ๐๐ญ. )(๐๐ ๐๐ญ. ) ๐ฌ๐ข๐ง ๐๐
๐๐ซ๐๐(๐จ๐ฉ๐ช) = (๐๐ ๐ฌ๐ข๐ง ๐๐) ๐๐ญ๐
๐๐ซ๐๐(๐จ๐ฉ๐ช) โ ๐๐ ๐๐ญ๐
7. A right rectangular pyramid has a square base with sides of length ๐. Each lateral face of the pyramid is an isosceles
triangle. The angle on each lateral face between the base of the triangle and the adjacent edge is ๐๐ยฐ. Find the
surface area of the pyramid to the nearest tenth.
Using tangent, the altitude of the triangle to the base of length ๐ is equal to ๐. ๐ ๐ญ๐๐ง ๐๐.
๐๐ซ๐๐ =๐
๐๐๐
๐๐ซ๐๐ =๐
๐(๐)(๐. ๐ ๐ฌ๐ข๐ง ๐๐)
๐๐ซ๐๐ = ๐. ๐๐(๐ฌ๐ข๐ง ๐๐)
The total surface area of the pyramid is the sum of the four lateral faces and the area of the
square base:
๐๐ = ๐(๐. ๐๐(๐ฌ๐ข๐ง ๐๐)) + ๐๐
๐๐ = ๐๐ ๐ฌ๐ข๐ง ๐๐ + ๐๐
๐๐ โ ๐๐. ๐
The surface area of the right rectangular pyramid is approximately ๐๐. ๐ square units.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area
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8. The Pentagon building in Washington, DC, is built in the shape of a regular pentagon. Each side of the pentagon
measures ๐๐๐ ๐๐ญ. in length. The building has a pentagonal courtyard with the same center. Each wall of the center
courtyard has a length of ๐๐๐ ๐๐ญ. What is the approximate area of the roof of the Pentagon building?
Let ๐จ๐ represent the area within the outer perimeter of the
Pentagon building in square feet.
๐จ๐ =๐๐๐
๐ ๐ญ๐๐ง (๐๐๐
๐)
๐จ๐ =๐ โ (๐๐๐)๐
๐ ๐ญ๐๐ง (๐๐๐
๐)
๐จ๐ =๐โ๐๐๐โ๐๐๐
๐ ๐ญ๐๐ง(๐๐)โ ๐โ๐๐๐โ๐๐๐
The area within the outer perimeter of the Pentagon building is
approximately ๐, ๐๐๐, ๐๐๐ ๐๐ญ๐.
Let ๐จ๐ represent the area within the perimeter of the
courtyard of the Pentagon building in square feet.
๐จ๐ =๐๐๐
๐ ๐ญ๐๐ง ๐๐
๐จ๐ =๐(๐๐๐)๐
๐ ๐ญ๐๐ง ๐๐
๐จ๐ =๐๐๐โ๐๐๐
๐ ๐ญ๐๐ง ๐๐
๐จ๐ =๐๐๐โ๐๐๐
๐ญ๐๐ง ๐๐โ ๐๐๐โ๐๐๐
Let ๐จ๐ป represent the total area of the roof of the
Pentagon building in square feet.
๐จ๐ป = ๐จ๐ โ ๐จ๐
๐จ๐ป =๐โ๐๐๐โ๐๐๐
๐ ๐ญ๐๐ง(๐๐)โ
๐๐๐โ๐๐๐
๐ญ๐๐ง ๐๐
๐จ๐ป =๐โ๐๐๐โ๐๐๐
๐ ๐ญ๐๐ง ๐๐โ
๐๐๐โ๐๐๐
๐ ๐ญ๐๐ง ๐๐
๐จ๐ป =๐โ๐๐๐โ๐๐๐
๐ ๐ญ๐๐ง ๐๐โ ๐โ๐๐๐โ๐๐๐
The area of the roof of the Pentagon building is approximately ๐, ๐๐๐, ๐๐๐ ๐๐ญ๐.
9. A regular hexagon is inscribed in a circle with a radius of ๐. Find the perimeter and area of the hexagon.
The regular hexagon can be divided into six equilateral triangular regions with
each side of the triangles having a length of ๐. To find the perimeter of the
hexagon, solve the following:
๐ โ ๐ = ๐๐ , so the perimeter of the hexagon is ๐๐ units.
To find the area of one equilateral triangle:
๐๐ซ๐๐ =๐
๐(๐)(๐) ๐ฌ๐ข๐ง ๐๐
๐๐ซ๐๐ =๐๐
๐(
โ๐
๐)
๐๐ซ๐๐ =๐๐โ๐
๐
The area of the hexagon is six times the area of the equilateral triangle.
๐๐จ๐ญ๐๐ฅ ๐๐ซ๐๐ = ๐ (๐๐โ๐
๐)
๐๐จ๐ญ๐๐ฅ ๐๐ซ๐๐ =๐๐๐โ๐
๐โ ๐๐๐. ๐
The total area of the regular hexagon is approximately ๐๐๐. ๐ square units.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 31 GEOMETRY
Lesson 31: Using Trigonometry to Determine Area
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10. In the figure below, โ ๐จ๐ฌ๐ฉ is acute. Show that ๐๐ซ๐๐ (โณ ๐จ๐ฉ๐ช) =๐๐
๐จ๐ช โ ๐ฉ๐ฌ โ ๐ฌ๐ข๐ง โ ๐จ๐ฌ๐ฉ.
Let ๐ฝ represent the degree measure of angle ๐จ๐ฌ๐ฉ, and let ๐ represent the
altitude of โณ ๐จ๐ฉ๐ช (and โณ ๐จ๐ฉ๐ฌ).
๐๐ซ๐๐(โณ ๐จ๐ฉ๐ช) =๐๐
โ ๐จ๐ช โ ๐
๐ฌ๐ข๐ง ๐ฝ =๐
๐ฉ๐ฌ, which implies that ๐ = ๐ฉ๐ฌ โ ๐ฌ๐ข๐ง ๐ฝ.
Therefore, by substitution:
๐๐ซ๐๐(โณ ๐จ๐ฉ๐ช) =๐๐
๐จ๐ช โ ๐ฉ๐ฌ โ ๐ฌ๐ข๐ง โ ๐จ๐ฌ๐ฉ.
11. Let ๐จ๐ฉ๐ช๐ซ be a quadrilateral. Let ๐ be the measure of the acute angle formed by diagonals ๐จ๐ชฬ ฬ ฬ ฬ and ๐ฉ๐ซฬ ฬ ฬ ฬ ฬ . Show that
๐๐ซ๐๐(๐จ๐ฉ๐ช๐ซ) =๐๐
๐จ๐ช โ ๐ฉ๐ซ โ ๐ฌ๐ข๐ง ๐.
(Hint: Apply the result from Problem 10 to โณ ๐จ๐ฉ๐ช and โณ ๐จ๐ช๐ซ.)
Let the intersection of ๐จ๐ชฬ ฬ ฬ ฬ and ๐ฉ๐ซฬ ฬ ฬ ฬ ฬ be called point ๐ท.
Using the results from Problem 10, solve the following:
๐๐ซ๐๐(โณ ๐จ๐ฉ๐ช) =๐๐
๐จ๐ช โ ๐ฉ๐ท โ ๐ฌ๐ข๐ง ๐ and
๐๐ซ๐๐(โณ ๐จ๐ซ๐ช) =๐๐
๐จ๐ช โ ๐ท๐ซ โ ๐ฌ๐ข๐ง ๐
๐๐ซ๐๐(๐จ๐ฉ๐ช๐ซ) = (๐๐
๐จ๐ช โ ๐ฉ๐ท โ ๐ฌ๐ข๐ง ๐) + (๐๐
๐จ๐ช โ ๐ท๐ซ โ ๐ฌ๐ข๐ง ๐) Area is additive.
๐๐ซ๐๐(๐จ๐ฉ๐ช๐ซ) = (๐๐
๐จ๐ช โ ๐ฌ๐ข๐ง ๐) โ (๐ฉ๐ท + ๐ท๐ซ) Distributive property
๐๐ซ๐๐(๐จ๐ฉ๐ช๐ซ) = (๐๐
๐จ๐ช โ ๐ฌ๐ข๐ง ๐) โ (๐ฉ๐ซ) Distance is additive
And commutative addition gives us ๐๐ซ๐๐(๐จ๐ฉ๐ช๐ซ) =๐๐
โ ๐จ๐ช โ ๐ฉ๐ซ โ ๐ฌ๐ข๐ง ๐.