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Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa translated and adapted by Will Harte Copyright © 1995 Drofa Publishers Copyright © 2003 Perpendicular Press "It has an interesting approach, and is clearly a valuable pedagogical contribution." -Robert Megginson, Deputy Director, Mathematical Sciences Research Institute Price: $38.50 (including FREE shipping to anywhere in the U.S.)

4.7  Circles

Figure 4.5 below shows a drawing implement that you will be familiar with: the compass. At the end of one of its legs is a sharp point. At the other end, a pencil. If you put the sharp end on a piece of paper and rotate the compass, the pencil will make the circumference of a circle (fig. 4.6). The point on which the sharp end sits is called the center of the circle. The center of the circle is usually denoted by the letter O. All points on the circumference lie in a single plane and are an equal distance from the center.

fig. 4.5	fig.  4.6	fig.  4.7

 
 

Let's connect the center to an arbitrary point A on the circumference with a line segment (fig. 4.7). The resulting segment AO, as well as its length, is called the radius of the circle. We denote the radius by the letter r. All radii of the same circle are equal to each other. Now, let's draw a segment through the center connecting 2 points on the circumference (fig. 4.8). The resulting segment AB, as well as the length of the segment, is called the diameter of the circle. We denote the diameter by the letter d. The diameter is 2 times longer than the radius. With the help of a formula, this may be described in the following manner: d=2r.

fig. 4.8

The circumference of a circle divides any plane into 2 sections. The portion of the plane on the circle's circumference and inside of it forms a disk (fig. 4.9). The center, radius, and diameter of the circle are simultaneously the center, radius, and diameter of the disk. The distance from the center to any point on the disk cannot be greater than the radius.

522.	Read the text to make sure you understand:
	1. How points on the circumference of the circle are located in relation to the center;
	2. What radius and diameter means, and how they are related to each other;
	3. How circumference differs from a circle.

fig. 4.9	fig. 4.10

523.	Draw a circle and measure its radius and diameter.
524.	Find the diameter of a circle if its radius is: 4 cm.; 6.8 cm.; 42 cm.; 71 cm.; 0.6 m.; 30.84 m.; 1/4 m.; 2/5 m.
525.	Find the radius of a circle if its diameter is: 6.8 cm.;18 cm.;53 cm.;71/2 cm.;1 m.;27/10 m.;42.6 m.;61/2 m.
526.	Draw a circle with a radius of 5.2 cm.  Identify its center and show two of its diameters.
527.	From fig. 4.10, identify which points belong to:  a) the circumference; b) the circle; and c) the region outside the disk.

528.	Draw a circle with a radius of 3.4 cm.  Then draw any two radii that form a right angle.
529.	Draw a circle of any radius.  Mark on the circumference points A, B, and C such that the segment AB forms the diameter of the circle.  Draw segments AC and BC.  Measure angle ACB.
530.	The center of a circle is the point O, the length of its radius r.  Where is the point P located if:  a) OP<r; b) OP= r; and c) OP>r?
531.	Extra Credit:  Draw a circle and a rectangle such that they share:  a) a single common point; b) 2 common points;  c) 3 common points; and d) 4 common points.

4.8:  Circumference

Problem:	A circular flower bed has a diameter of 5 meters.  You need to surround the bed with a  strip of turf.  How long does your strip of turf need to be if you measure along its inside edge?
	In order to solve this problem, it is necessary to find the circumference of the circle.
	We can attempt to do this by actually measuring the circle, for instance with a “stepping” compass (fig.4.11).  But, of course, the result would not be exact, since we would have substituted a broken line for the circumference.

  

fig. 4.11	fig. 4.12

Instead, let’s try the following experiment:  Take a jar or a can and run a string around the outside of it.  If we then straighten out the string, its length will be approximately equal to the circumference of the jar.  In order to obtain a more precise result, we need to run the string around the jar several times and then take the entire length of the string and divide it by the number of times we went around the jar.

Next, we may measure the diameter of the jar with a tape or a try-square.  Let’s see how much larger the jar’s circumference is compared to its diameter.  In order to do this, we divide the circumference by the diameter.  If we have measured correctly, then the quotient will be somewhere between 3.1 and 3.2.  Exact mathematical models (which you will study later in school) demonstrate that the circumference of any circle is always larger than its diameter by one and the same number.  This number is denoted by the Greek letter p (Pi), whose exact value is expressed by the infinite, non-repeating decimal:

p = 3.14159265… .

Henceforth we will take an approximate value of p»3.14.  If we denote the circumference by the letter C, and its diameter d, then we get the formula for determining circumference:

C=  pd

The circumference of a circle is equal to the product of the number p and the circle’s diameter.  We know that d=2r, therefore the formula for determining circumference may be rewritten:

C=2pr.

Now we can solve the problem given at the beginning of this section.  Since the diameter  of the flower bed is 5 m., we may write:  C =p·5 » 3.14·5 = 15.7 (m.).  This means we must prepare a strip of turf approximately 15.7 m. long.

Many calculators have a special button for the number p.  If you press it, then an approximation of p will appear on the screen, usually to the seventh decimal place:

3.1415926

Example:  If the diameter of a circle is 3.8 m., then we may find the circumference on a calculator in the following manner:

The screen will display the answer 11.938051, which may be rounded to the tenth’s place (11.9 m.), the hundredth’s (11.94 m.), etc..

A.

532.	Answer the following questions with the help of the preceding text: a. Why can’t we measure the circumference of a circle with a ruler? b. How can we measure the circumference of  any object, for instance a glass? c. What does the letter p signify? d. How do you calculate the circumference of a circle?
533.	At home, measure the diameter and circumference of two different glasses.  Calculate the quotients from dividing the circumferences of the circles by their respective diameters.  (Round answers to the tenth’s place.)
534.	Calculate the circumference of a circle if its diameter is:  5.8 cm., 42 cm., 97 cm., 0.54 m., 43 m.
535.	Calculate the circumference of a circle if its radius is:  6 cm., 8.5 cm., 0.8 m., 46 cm., 0.35 m., 24 m.
536.	Measure the diameter of the circle in figure 4.12 and calculate its circumference.

  

537. 538.

Some boys and girls decided to organize a bicycle race after school.  The race was to consist of 4 laps around a small track with a radius of 3 meters (at right).  What distance did the boys and girls travel on their bikes? (Round your answer to the nearest whole number.) In order to decorate a room for New Year’s Eve, Annie cut out circles with an 8-cm. radius from colored paper.  How far did she have to go with her scissors in order to cut out 30 such circles? (Round your answer to the nearest whole number.)

B.

 

539.	What is the diameter of a circle if its circumference is 13.4 cm?
540.	What is the radius of a circle if its circumference is 56.32 m?
541.	Compare the perimeter of a square whose sides are 9 cm. to the circumference of a circle with a diameter 9 cm.

     

542.	A semicircle has been cut out of a rectangle (fig. 4.13).  Make the necessary measurements and calculate the perimeter of the remaining solid.	fig. 4.13
543.	In his science class, Andy built a model plane and took it out for a test run.  If the plane flew a circle with a 60-m. radius in 1.8 seconds, at what speed was the plane flying?
544.	In ancient Babylon, people calculated the circumference of a circle taking p to be equal exactly to three. Suppose the Babylonians had calculated the circumference of a circle with a radius of 40 feet. How far would their answer be from that which we in sixth grade get using p»3.14?
545.	The diameter of a circle is increased by 1 in.  By how much does the circumference then increase?

   

Problem:	A circle with a radius of 2 meters must be painted on the floor of a gym.  How much paint is needed if you need 0.2 kilograms of paint per square meter?
	In order to solve this problem, we have to find the area of the circle.  How do we do this?

 

fig. 4.14

Stringent mathematical reasoning has proven that the area of a circle is p times the area of a square whose side is equal to the radius of the circle.  If we denote the radius of the circle by the letter r, then the area of a square whose side is equal to the radius is r2.  It follows that the formula for the area of a circle is calculated by the formula (fig. 4.14):

A=pr2.

In order to calculate the area of a circle, you must multiply the square of the radius by p. With the help of this rule we may solve the problem formulated at the start of this section.  We can find the area of the circle we need to paint by the formula:

A= 3.14·2 = 12.56 (m2).

Since for every square meter we need 0.2 kilograms of paint, the total amount of paint we will need to buy is 0.2·12.56 » 2.5 (kg.).

In order to find the area of the circle using a calculator, you need to know where how to square numbers, that is, how to multiply a number by itself.  This is very simple:  we select a number, press the multiply sign and then the equal sign.  The screen will now show the square of the number.

EXAMPLE 1: Calculate 42.
Procedure:
EXAMPLE 2: Calculate the area of a circle with a radius of 3.7 cm.
Procedure:
The screen will show us the number 43.008402, which if necessary can be rounded, for instance, to the tenth's place (43.0 cm2), to the hundredth's place (43.01 cm2), etc..

A Little History

Even in ancient times people were familiar with many geometric shapes, including the circle.  This is evident from archaeological diggings which have turned up various ornaments, china, and remains of old weapons.  This means that even then people were calculating the circumference and area of circles.  We now know that p has been represented by different numbers throughout the ages.  Over 3,500 years ago in Egypt, for instance, p was taken to be 3.16, while the ancient Romans believed p = 3.12.  All these values were discovered through experiments. The great scholar of ancient Greece, Archimedes (287-212 BC), calculated the value of p as being between 310/71< p < 31/7, or 3.1408…< p <3.1428… .  With the help of modern computers, p has now been calculated to over 2 billion decimal places.  The British mathematician William Jones in 1706 was the first person to use the letter p to denote the quotient obtained dividing circumference by diameter.  But this denotation became popular thanks to the work of the great mathematician Leonhard Euler (1707-1783), who was a member of the St. Petersburg Academy of Sciences.  He calculated p to 153 decimal places.

    

546.	What is the main point of the preceding paragraph?  How are the area of a circle and the area of a square related to each other?
547. 548.  549. 550.	Calculate the area of a circle if its radius is 6 cm., 4.2 cm., 1.2 m., 3.4 m., 23 m.. Calculate the area of a circle if its diameter is 18 cm., 36cm., 1.5 m., 2 m., 8.5 m.. Measure the radius of the circle in figure 4.10, and calculate its area and circumference. A circular playground with a radius of 3.5 m. has been set up around a flagpole. Calculate the area of the playground. (Round your answer to the tenth’s place.)
551.	As part of  rebuilding a three-story round  tower, all floors were covered with tile. How many square meters of tile were needed if the inner diameter of the tower was 6.4 m.? (Round your answer to the nearest whole number.)

  

552.	The radius of one of the circles pictured in figure 4.15 is 12 cm., while the radius of the other is 3 cm. greater. Calculate the area of the ring formed by these two circles.

 

fig. 4.15	fig. 4.16	fig. 4.17

553.	Five round flower beds, each with a diameter of 1.5 m. are laid out in a park.  Nine flowers per square meter should be planted in the beds.  Formulate a question involving this problem and solve it.
554.	Extra Credit:  A semicircle is placed on each side of a square, as in fig. 4.16.  Make the necessary measurements and calculate the area of the resulting shape.
555.	Extra Credit:  The sides of the square shown in fig. 4.17 are each 4 cm.  What is the radius of the circle? How many times larger than the circle is the area of the square?  (Round your answer to the hundredth's place.)

  

556.	Calculate and draw the following angles:  1)  30% of 180º; 2) 60% of 70º; 3) 45% of 160º.
557.	Calculate (Round your answers to a whole number):
	1)  74% of 130º,       6% of 78º,       4/9 of 160º; 2) 92% of 90º,      8.4% of 180º,      0.24 of 88º. 3) 54% of 165º,      2.3% of 136º,      2/7 of 36º.
558.	Draw angles measuring 35º and 80º such that their vertices are the same and they have a common side.  Provide two examples.
559.	On a 6th-grade math test 25% of the class earned an A, 35% a B, 30% a C, and 10% a D. Draw a bar graph depicting the grade distribution.
560.	Calculate: 1) 0.75 ÷ 5/6 + 2.5 · 2/5 - 1 ÷ 11/9 2) ((45/12 - 313/24) · 4/7 + 110/17 (31/18 - 27/12)) ÷ 31/3
561.	The famous Greek mathematician Archimedes determined that 310/71< p < 31/7. Compare the circumference of a circle using 310/71 and 31/7 as p if the circle's radius is 497 cm..