Applications of Radian
Measure
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C ONCEPT
1
1
Applications of Radian
Measure
Learning Objectives
A student will be able to:
•
•
•
•
Solve problems involving angles of rotation using radian measure.
Solve problems by calculating the length of an arc.
Solve problems by calculating the area of a sector.
Approximate the length of a chord given the central angle and radius.
Introduction
In this lesson students will apply radian measure to various problem-solving contexts involving rotations.
Rotations
Example 1
The hands of a clock show 11:20. Express the obtuse angle formed by the hour and minute hands in radian measure
to the nearest tenth of a radian.
The following diagram shows the location of the hands at the specified time.
π
◦
Because there are 12 increments on a clock, the angle between each hour marking on the clock is 2π
12 = 6 (or 30 ).
π
2π
◦
So, the angle between the 12 and the 4 is 4 × 6 = 3 (or 120 ). Because the rotation from 12 to 4 is one-third of a
complete rotation, it seems reasonable to assume that the hour hand is moving continuously and has therefore moved
π
one-third of the distance between the 11 and the 12. So, 13 × π6 = 18
, and the total measure of the angle is therefore
π
2π
π
12π
13π
+
=
+
=
.
Using
a
calculator
to
approximate
the
angle
would give:
18
3
18
18
18
CONCEPT 1. APPLICATIONS OF RADIAN MEASURE
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To the nearest tenth of a radian it is 2.3 radians.
Length of Arc
The length of an arc on a circle depends on both the angle of rotation and the radius length of the circle. If you recall
from the last lesson, we defined a radian as the length of the arc the measure of an angle θ in radians is defined as
the length of the arc cut off by one radius length, so that a half-rotation is π radians, or a little more than 3 radius
lengths around the circle. What if the radius is 4 cm? The length of the half-circle arc would be π radius lengths, or
4π cm in length.
This results in a formula that can be used to calculate the length of any arc.
s = rθ,
where s is the length of the arc, r is the radius, and θ is the measure of the angle in radians.
Solving this equation for θ will give us a formula for finding the radian measure given the arc length and the radius
length:
θ=
s
r
Example 2
The free-throw line on an NCAA basketball court is 12 ft wide. In international competition, it is only about 11.81 ft.
How much longer is the half circle above the free-throw line on the NCAA court?
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Arc Length Calculations:
NCAA
INTERNATIONAL
s1 = rθ
s2 = rθ
s1 = 12(π)
s2 ≈ 11.81(π)
s1 = 12π
s2 ≈ 11.81π
So the answer is approximately 12π − 11.81π ≈ 0.19π
This is approximately 0.6 ft, or about 7.2 inches longer.
Example 3
Two connected gears are rotating. The smaller gear has a radius of 4 inches and the larger gear’s radius is 7 inches.
What is the angle through which the larger gear has rotated when the smaller gear has made one complete rotation?
CONCEPT 1. APPLICATIONS OF RADIAN MEASURE
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Because the blue gear performs one complete rotation, the length of the arc traveled is:
s = rθ
s = 4 × 2π
So, an 8π arc length on the larger circle would form an angle as follows:
s
r
8π
θ=
7
θ ≈ 3.6
θ=
So the angle is approximately 3.6 radians.
◦
3.6 × 180
π ≈ 206
Area of a Sector
One of the most common geometric formulas is the area of a circle:
A = πr2
In terms of angle rotation, this is the area created by 2π radians.
2π radian angle = πr2 area
A half-circle, or π radian rotation would create a section, or sector of the circle equal to half the area or:
1 2
πr
2
So an angle of 1 radian would define an area of a sector equal to:
2Z
π Z
πr2
=
2Z
π
2Z
π
1 2
1= r
2
From this we can determine the area of the sector created by any angle, θ radians, to be:
1
A = r2 θ
2
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Example 4
Crops are often grown using a technique called center pivot irrigation that results in circular shaped fields.
FIGURE 1.1
Here is a satellite image taken over fields in Kansas that use this type of irrigation system. You can read more about
this at: http://en.wikipedia.org/wiki/Center_pivot_irrigation
FIGURE 1.2
If the irrigation pipe is 450 m in length, what is the area that can be irrigated after a rotation of
2π
3
radians?
CONCEPT 1. APPLICATIONS OF RADIAN MEASURE
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Using the formula:
1
A = r2 θ
2
1
2 2π
A = (450)
2
3
The area is approximately 212, 058 square meters.
Length of a Chord
You may recall from your Geometry studies that a chord is a segment that begins and ends on a circle.
AB is a chord in the circle.
We can calculate the length of any chord if we know the angle measure and the length of the radius. Because each
endpoint of the chord is on the circle, the distance from the center to A and B is the same as the radius length.
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Next, if we bisect angle, the angle bisector must be perpendicular to the chord and bisect it (we will leave the proof
of this to your Geometry class!). This forms a right triangle.
We can now use a simple sine ratio to find half the chord, called c here, and double the result to find the length of
the chord.
θ
c
sin
=
2
r
θ
c = r × sin
2
So the length of the chord is:
CONCEPT 1. APPLICATIONS OF RADIAN MEASURE
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θ
2c = 2r sin
2
Example 5
Find the length of the chord of a circle with radius 8 cm and a central angle of 110◦ . Approximate your answer to
the nearest mm.
It’s always a good problem solving technique to estimate the answer first. A thought process for estimating the
measure might look something like this:
The angle is slightly more than a 90◦ , or π2 radians. π2 radians is slightly more than 1.5 radius lengths. One and a half
radii would be 12, so we might expect the answer to be a little more than 12 cm. Let’s see how the actual answer
compares.
We must first convert the angle measure to radians:
110 ×
π
11π
=
180
18
Using the formula, half of the chord length should be the radius of the circle times the sine of half the angle.
11π 1 11π
× =
18
2
36
8 × sin
11π
36
(Make sure your calculator is in radians!!!)
Multiply this result by 2.
So, the length of the arc is approximately 13.1 cm. This seems very reasonable based on our estimate.
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Further Reading
• http://en.wikipedia.org/wiki/Basketball_court
• http://en.wikipedia.org/wiki/Center_pivot_irrigation
• http://www.colorado.gov/dpa/doit/archives/history/symbemb.htm#Flag
Review Questions
1. The following image shows a 24 − hour clock in Curitiba, Paraná, Brasil.
FIGURE 1.3
a. What is the angle between each number of the clock expressed in:
a. exact radian measure in terms of π ?
b. to the nearest tenth of a radian?
c. in degree measure?
b. Estimate the measure of the angle between the hands at the time shown in:
a. to the nearest whole degree
b. in radian measure in terms of π
2. The following picture is a window of a building on the campus of Princeton University in Princeton, New
Jersey.
CONCEPT 1. APPLICATIONS OF RADIAN MEASURE
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FIGURE 1.4
(a) What is the exact radian measure in terms of π between two consecutive circular dots on the small circle
in the center of the window?
(b) If the radius of this circle is about 0.5 m, what is the length of the arc between the centers of each
consecutive dot? Round your answer to the nearest cm.
3. Now look at the next larger circle in the window.
a. Find the exact radian measure in terms of π between two consecutive dots in this window.
b. The radius of the glass portion of this window is approximately 1.20 m. Calculate an estimate of the
length of the highlighted chord to the nearest cm. Explain the reasoning behind your solution.
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4. The state championship game is to be held at Ray Diaz Memorial Arena. The seating forms a perfect circle
around the court. The principal of Archimedes High School is sent the following diagram showing the seating
allotted to the students at her school.
a. the students from Archimedes.
b. general admission.
c. the press and officials.
It is 55 ft from the center of the court to the beginning of the stands and 110 ft from the center to the end. Calculate
the approximate number of square feet each of the following groups has been granted:
5. This is an image of the state flag of Colorado
FIGURE 1.5
The detailed description of the proportions of the flag can be found at: http://www.colorado.gov/dpa/doit/archives/hi
story/symbemb.htm#Flag
It turns out that the diameter of the gold circle is 13 the total height of the flag (the same width as the yellow stripe)
and the outer diameter of the red circle is 23 of the total height of the flag. The angle formed by the missing portion
of the red band is π4 radians. In a flag that is 33 inches tall, what is the area of the red portion of the flag to the nearest
square inch?
CONCEPT 1. APPLICATIONS OF RADIAN MEASURE
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Review Answers
1.
π
a. 12
b. ≈ 0.3 radians
c. 15◦
a. 20◦ . Answers may vary, anything above 15◦ and less than 25◦ is reasonable.
b. π9 Again, answers may vary
a. π/6
b. ≈ 26 cm
a. π/6
b. Let’s assume, to simplify, that the chord stretches to the center of each of the dots. We need to find the
measure of the central angle of the circle that connects those two dots.
Since there are 13 dots, this angle is
13π
16 .
The length of the chord then is:
θ
=2r sin
2
1 13π
=2 × 1.2 × sin
×
2
16
The chord is approximately 2.30 cm.
2. Each section is π6 radians. The area of one section of the stands is therefore the area of the outer sector minus
the area of the inner sector:
a. The students have 4 sections or ≈ 9503 ft2
b. There are 3 general admission sections or ≈ 7127 ft2
c. There is only one press and officials section or ≈ 2376 ft2
A = Aouter − Ainner
π 1
π
1
A = (router )2 × − (rinner )2 ×
2
6 2
6
1
π 1
π
2
2
A = (110) × − (55) ×
2
6 2
6
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The area of each section is approximately 2376 ft2 .
3. There are many difference approaches to the problem. Here is one possibility: First, calculate the area of the
red ring as if it went completely around the circle:
A = Atotal − Agold
2
2
2
1
A=π
× 33 − π
× 33
3
3
A = π × 222 − π × 112
A = 484π − 121π = 363π
A ≈ 1140.4 in2
Next, calculate the area of the total sector that would form the opening of the “c”
1
A = r2 θ
2
π
1
A = (22)2
2
4
A ≈ 190.1 in2
Then, calculate the area of the yellow sector and subtract it from the previous answer.
1
A = r2 θ
2
π
1
A = (11)2
2
4
A ≈ 47.5 in2
190.1 − 47.5 = 142.6 in2
Finally, subtract this answer from the first area calculated.
CONCEPT 1. APPLICATIONS OF RADIAN MEASURE
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The area is approximately 998 in2
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CONCEPT 1. APPLICATIONS OF RADIAN MEASURE