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Mathematics 6

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.






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: 

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. 




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. 
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. 




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 trysquare. 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, nonrepeating
decimal: 



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: 



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: 



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: 



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.. 
532. 
Answer the following
questions with the help of the preceding text: 
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. 
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.) 

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 60m. 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? 


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 r^{2}. It follows
that the formula for the area of a circle is calculated by the formula
(fig. 4.14): 

In order to
calculate the area of a circle, you must multiply the square of the
radius by p. 

Since for every
square meter we need 0.2 kilograms of paint, the total amount of paint
we will need to buy is 
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 4^{2}. 

Procedure: 



Procedure: 



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. 
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. 
Calculate the area of
a circle if its radius is
6 cm., 4.2 cm., 1.2 m., 3.4 m., 23 m.. 

551. 
As part of
rebuilding a threestory 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. 






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): 



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 6thgrade math
test 25% of the class earned an A, 35% a B, 30% a C, and 10% a D. 

560. 
Calculate: 

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.. 
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