ARC MEASURMENT BASED ON THE MEASURMENT OF A CENTRAL ANGLE

Keywords:
Central angle –An angle whose vertex is the center of a circle and
whose sides contain the radii of the circle

Arc – Two points on a circle and the continuous (unbroken) part of
the circle between the twp points. The two points are called endpoints

Degrees – A unit of measure for angles

Radians – Is an Standard International (SI) unit of plane angular
measurement

Radius – A segment from a point on the sphere of circle to the
center. The length of the segment is also called radius

Objectives:
1) Students will be able to determine the length of an arc
of a circle given the central angle.
2) Student will be able to convert from radians to degrees.
3) Students will be able to use a macro.

Materials:
1) Cabri or Geometer Sketch Pad
2) Pencil and paper
3) Lab handout

Ohio Standards:
1) Analyze characteristics and properties of two - and three – dimensional geometric
shapes and develop mathematical arguments about geometric relationships p 310
2) Specify locations and describe special relationships. P 313
3) Use visualization, special reasoning and geometric modeling to solve problem. P 315

ARC MEASURMENT BASED ON THE MEASURMENT OF A CENTRAL ANGLE
Team Members:
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File name:

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This lab was de signed with the assumption that students have working knowledge Cabri
and/or Geometer Sketch Pad.
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Question: Given a circle, central angle, and the radius, can the length of a
corresponding arc be determined in degrees?

Students should already have general knowledge of circles. For instance, students should already know
that there are:
1) 360 degrees in a circle and,
2) 2(pi)(r) = 360 degrees
(where Pi = 3.14 and r = radians or radius).

Setting the radius of a circle r = 1 simplifies the equations that makes it clear that:
1) 1 radian = 180/Pi degrees and,
2) 1 degree = Pi/180

Understanding this relationship al lows us to measure the length of arc in degrees, given the circle, central
angle, and radius. The following is an illust ration of this property .

TASKS:

If you are using Cabri, notice that the arc measurement is given in centimeters. We will use our
knowledge of circles to convert this measurement from centimeters to degrees. The following steps
will guide you through this process.

1) Take the arc measurement (in centimeters)
Divide it by the measurement (in centimeters) of the radius.

(centimeters) / Radius = ________arc (radians)

( calculator tool )

Drag you results to your workspace and label (radians)
(Note: 1 Radian = 180 degrees/Pi)

2) Take the arc measurement (in radians) and multiply it by 360 degrees. Then, divide that
product by (2* Pi).

(radians) * 360 / (2 * Pi) = __________arc (degrees)

(calculator tool)

(radians) * 180 / Pi = ___________arc (degrees)
Drag the result from the calculator tool and place it by the arc. Use comment tool to
rename in degrees.

Now you are capable of converting arc measurements in centimeters to an arc measurement in
degrees. With this knowledge, you can create a macro. To create the macro:

1) Click on the macro tool.
2) Select Initial Object.
3) Then click on the complete circle then click on points A, C, and B
4) Click on macro tool
5) Select Final Object
6) Click on number value of the arc measurement in degrees.
7) Click on macro tool and select Define Macro
8) Fill in the “Name of the construction”
Call it - Arc measurement in degrees
9) Fill in “Name for final object”
Call it - Arc measurement in degrees
10) Fill in the “Help for this macro”
Write - Given a circle, radius and an arc defined by three points on the circle, this macro
will calculate the arc in degrees

Now you can use your new macro to investigate the relationship between an arc of a circle and the
corresponding central angle.

1) Draw a circle	(circle tool)
2) Draw the diameter of the circle	(line tool)
3) Draw a perpendicular line to the diameter passing through the center of the circle	(perpendicular line tool)
4) Draw all four points of intersection	(point tool)
5) Label the points of intersection ABCD and label the center O.	(label tool)
6) Make a point on the circle P between point A and point B	(point tool)
7) Place an arc from point A to point P and point B	(arc tool)
8) Measure = ______	(angle tool)
9) Use your macro to measure the	(macro)

Using the same circle, construct, measure, and label five central angles. Then, use your macro to
measure the corresponding arc.

With your expanded knowledge, what is the measure of∠BOD? = ______
Also, with your macro, what is the measure of the corresponding arc? = ______

In radians, what is the measurement of this arc? _______

Conclusions:

Write you conjecture here. What can you say about any central angle of a circle and its’
corresponding arc?
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Extensions:

Your math teacher decided to have a pizza party in your honor. The teacher announces that any
student interested in a piece of pizza must first answer a challenging geometry question.

1) Suppose you were really hungry . Without looking at the 16” pizza which would you prefer ?

a. A piece of pizza that had a corresponding arc = 5 radians
b. A piece of pizza that has a corresponding arc = 287 degrees
c. A piece of pizza that has an arc measurement of 45 inches