Transcript Using the Inquiry Page in a High School Classroom

How to find the area of a circle
from segments
Anuradha Datta Murphy
University of Illinois at Urbana-Champaign

Outline of presentation:
–
–
–
–
–
–
–
Concept of area
Area expressed as mathematical formulae
Definition of a circle and its components
Concept of p
How to find the area of a circle
Show Wolfram Demonstrations tutorial
Explanation of how to find the area of a circle from
its segments
– Suggestion of further exercises in area
calculations
What is area?

Area is defined as the size of a twodimensional surface, typically enclosed
by a closed curve.
Areas expressed by mathematical
formulas:

Areas of figures can be calculated using
mathematical formulas. The following 2dimensional figures have simple formulas for
calculating their area:
Square: area = S2, Where S is the length of one side
S
Triangle: area = ½ bxh, where b is the length of the
base and h is the height
h
b
Parallelogram: area = bh, where b is the length
of any side, and h is the distance between the
lines of b
h
b
What is a circle?


A circle is defined as a curved line surrounding a
central point, every point on the line being equidistant
from the center.
In the figure below, the curve is equidistant from the
point o, the center of the circle.
o
b
r
a


The distance r is defined as the radius of the circle.
The section aob is called a segment of the circle.
What is p?



The distance around a circle is called its
circumference.
The distance across a circle through its radius (i.e.,
twice the radius) is called its diameter.
The Greek letter p is used to represent the ratio of
the circumference of a circle to its diameter.
d = 2r
d
.
Circumference, c = pd
p = c/d = 3.14
Area of a circle:

The area enclosed by a circle with radius r is
Area = pr2
Or, given a diameter d,
Area = p(d/2)2 = (pd2)/4
r
d
Viewing Wolfram Demonstration on how to
find area of a circle from segments:




Go to the Wolfram Demonstrations Project site at
http://demonstrations.wolfram.com/
Type “area” in the Search window
From the search results, select “Area of a Circle from
Segments” by clicking on it
Click on “watch web preview” to view a short
demonstration
Viewing the Wolfram Demonstration:
– In the demonstration, the circle is divided into many
segments. These segments are stacked on the side of the
circle.
segments
Viewing the Wolfram Demonstration (contd.):
– As the number of segments increases, the stack resembles
a rectangle. Since the circumference of the circle is 2pr, the
rectangle has sides r and pr.
Area of rectangle = pr x r = pr2
and, Area of circle = pr2
r
pr
Circumference = 2pr
Viewing the Wolfram Demonstration (contd.):



To view the Wolfram Demonstration interactively, click
on the “Download Live Version” button on the
demonstration screen
*Note*: Mathematica Player has to be installed on
your desktop before this application can be run
In the Live Version of the demonstration, you can
change the number of segments in the circle by
clicking on the slider. As you change the number of
segments, observe that increasing the number
makes the stack of segments resemble a rectangle,
which helps to illustrate the formula for calculating the
area of a circle.
Further exercises in area calculation:


Return to the area search results page.
Select at least three other shapes and view the
demonstrations on area calculations for each of
these. For example, you could try the following:
– Area of a Parallelogram
– Area of a Triangle as Half a Rectangle
– Maximizing the Surface Area of a Cylinder with a Fixed
Volume
Closing Notes:

For further illustrations on calculating the area of
circles, take a look at:
– http://www.easycalculation.com/area/learn-circle.php
– http://www.wikihow.com/Calculate-the-Area-of-a-Circle
– http://www.mathwarehouse.com/geometry/circle/area-of-circle.php

For more examples of simple area calculations, see:
– http://en.wikibooks.org/wiki/Geometry/Area
– http://www.webmath.com/index5.html
– http://www.kidsnewsroom.org/elmer/infocentral/geometry/Area.html