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Dividing A
Circle Into Thirds
How to Divide a Circle Into Thirds
By Kevin Beck
Circles
are everywhere in nature, art and the sciences.
The
sun and the moon, through spherical, form circles in the sky and travel in
roughly circular orbits; the hands of a clock and the wheels on automobiles
trace out circular paths; philosophically-minded observers speak of a the
"circle of life."
Circles in plain terms are mathematical constructs. You may
need to know, using math, how to separate a complete circle into equal portions
for pie, land or artistic purposes.
If
you have a pencil, along with a protractor, a compass or both, dividing a
circle into three equal parts is straightforward and instructive.
A circle encloses 360 degrees of an arc, so for this
exercise you need to create a "pie" with three equal 120° angles at
the center.
Step 1: Draw the Diameter
Use your straightedge (ruler or protractor) to draw a
diameter or line through the middle of the circle that reaches both edges. This
of course divides your circle in half.
Step
2: Mark the Center
If the center of the circle is not marked, you will find it
in this step because the diameter of any circle is the longest distance across
the circle.
Simply
divide the value of the diameter by 2 and place a point halfway along the line
from one edge to indicate the center.
Step 2: Measure Halfway to One Edge
Use your ruler or protractor to find a point exactly halfway
between the center and one edge, or equivalently, one-fourth of the diameter or
half of the radius. Label this point A.
Step 3: Draw a Perpendicular Line Through Point A to Both
Edges
Use your protractor, or if necessary the short edge of your
ruler, to draw a line through point A. Extend this line to the edges of the
circle.
Label
the points at which this line intersects the edge of the circle B and C.
Step 4: Draw Lines from the Center to Points B and C
Using your straightedge, create lines connecting the center
of the circle to points B and C. These lines represent radii of the circle,
which have a value of half of the diameter.
Step 5: Use Geometry to Solve the Problem
You now have two right triangles inscribed within the
circle.
Because
the short leg of each of these is one-half the distance of the hypotenuse of
the circle, which is the same as a radius, you may recognize that these right
triangles are "30-60-90" triangles, which have the property of the
shortest side being half the length of the longest.
Because of this, you can conclude that the interior angles
of the circle you have created between the two hypotenuses, and the hypotenuse
and the diameter on the opposite side of the circle, are each 120°.
You
thus have a circle divided into three equal parts.
Things Needed
·
Paper
·
Pencil
·
Compass
·
Ruler
About
the Author
Kevin Beck
holds a bachelor's degree in physics with minors in math and chemistry from the
University of Vermont. Formerly with ScienceBlogs.com and the editor of
"Run Strong," he has written for Runner's World, Men's Fitness,
Competitor, and a variety of other publications. More about Kevin and links to
his professional work can be found at www.kemibe.com.
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