Shape 

Copyright David & Cynthia Thomas, 2009

 

Circumference & Area of Circles--Concepts & Connections

 

Challenge: Measuring the Circumference of the Earth.  Every Spring, the Noon Day Project leads upper elementary, middle school and high school students (ages 11-18) through an investigation that imitates an experiment conducted by Eratosthenes (275-194 BC) over 2000 years ago.  Eratosthenes’ intent was to determine the circumference of the Earth measured along what we today call a line of longitude.  At the time, Eratosthenes was chief librarian in Alexandria, host city to the greatest library of the ancient world.  Like Aristotle and many other educated people of his time, Eratosthenes believed that the Earth was a sphere, the most direct evidence of this fact being the roundness of the Earth’s shadow on the moon during eclipses. What Eratosthenes wanted was a mathematical basis for estimating the Earth’s circumference.

 

One day, while reading a report from Syene (now called Aswan), a frontier town south of Alexandria, he learned that, at noon on June 21st, vertical sticks cast no shadow.  Eratosthenes realized that this observation provided a mathematical basis for a proving whether the Earth is flat or round.  His reasoning was simple yet powerful: If the Earth is flat, then parallel rays from the sun should cast identical shadows, (i.e., no shadow at all) at Alexandria and Syene on June 21st; but if the Earth is round, parallel rays should cast different shadows at the two locations (See Figure 3.34).

 

Flat Earth                                                      Round Earth

 

Figure 3.34: Is the Earth Flat or Round?

 

The geometric reasoning used by Eratosthenes is illustrated in Figures 3.35 and 3.36.   Eratosthenes knew that, in Figure 3.36, the arc length AB, the central angle ÐACB, and the circumference of the circle are related as follows:  (Why?).  When Eratosthenes’ realized that the central angle and the sun angle in Figure 3.36 were equal, he knew that he could calculate the circumference of the Earth (Why?).  All he needed to know was the sun angle and the distance from Alexandria to Syene.  Eratosthenes measured the sun’s angle as 7degrees12’.  Since the distance from Alexandria to Syene was known to be about 5000 stadia (Note: one stadia is approximately 157 meters) he obtained an answer of 250,000 stadia, a distance equivalent to 24,531 miles.

 

 

                                                                                      

Figure 3.35: Arc Length and Central Angles

 

Figure 3.36: Sun Angle = Central Angle

 

You may experiment with a Geometers Sketchpad model of Figure 3.36 using 1_SunAngle.gsp.

 

Directed Activity Diameter, Circumference, and Pi.  One of the most important discoveries of ancient mathematics was the relationship between the diameter and circumference of a circle: For every circle, the circumference divided by the diameter is a constant approximated by the decimal number 3.14159.  This number, identified worldwide by the Greek letter p, is known to be an irrational number (i.e., it cannot be expressed as the ratio of two integers, such as 22/7).  While its value has been computed to over ten billion decimal places, its utility in science and engineering rarely requires even a dozen decimal places.  In most applications, approximating p with the decimal 3.14159 produces acceptable results. 

 

The relationship itself,, is often the focus of student investigations in which the circumference and diameter of various cylinders are measured and the quotient computed.  In activities of this sort, errors in measurement invariably occur as students struggle to measure across the tube, not knowing for certain whether their line of measurement is, in fact, a diameter or if their measurement of the circumference is accurate.  For instance, Figure 3.37 shows a dot plot of 20 measurements of a cardboard tube with a diameter of 8cm and a circumference of 25.13cm.  A line of best fit is drawn through these points.  The slope of this line is 3.166, a good approximation of p.

Figure 3.37: Diameter vs. Circumference --- 20 Measurements

 

Directed Activity Diameter, Circumference, and Pi.  Figure 3.38 shows a Geometers Sketchpad model 3_CircleArea.gsp that illustrates the logical basis for the formula A = pr2, the area of a circle.  In this model, the circle is divided into pie-shaped sectors.  These sectors are then rearranged horizontally with alternating up and down orientations.  This arrangement vaguely resembles a parallelogram with scalloped top and bottom edges.  In the case of Figure 3.38, twelve 20° sectors cover roughly the same area as the shaded gray rectangle.  That rectangle has an area of 51.12 cm2, a reasonably good approximation of the true area of the circle, 53.47 cm2.  As thinner and thinner sectors are used, the height of the rectangle approaches r, the radius of the circle (Why?), and the length of the rectangle approaches half the circumference of the circle, pr (Why?).  Since the area of the rectangle is computed as (height)(length), its area may be approximated as (r)(pr), or pr2.   Imagining this process and its outcomes provides a strong intuitive justification for the formula A = pr2, a far better basis for mathematical success than memorization alone.

Figure 3.38: Area of a Circle

 

Example 3.3

Find the area of a circle with diameter 10cm. 

 

Solution 3.3

Since the diameter of the circle is 10cm, the radius is 5cm.  Then the area is computed as p(5)2 = 25p, or approximately 78.54cm2