Pi is an irrational number that cannot be written as a
repeating decimal or as a fraction. It has an infinite number of
non- repeating decimal places. Therefore, 3.5.1: Circumference and
Area of a Circle2
Slide 4
A limit is the value that a sequence approaches as a
calculation becomes more and more accurate. This limit cannot be
reached. Theoretically, if the polygon had an infinite number of
sides, could be calculated. This is the basis for the formula for
finding the circumference of a circle. 3.5.1: Circumference and
Area of a Circle3
Slide 5
The area of the circle can be derived similarly using
dissection principles. Dissection involves breaking a figure down
into its components. 3.5.1: Circumference and Area of a
Circle4
Slide 6
The circle in the diagram to the right has been divided into 16
equal sections. 3.5.1: Circumference and Area of a Circle5
Slide 7
You can arrange the 16 segments to form a new rectangle. This
figure looks more like a rectangle. 3.5.1: Circumference and Area
of a Circle6
Slide 8
As the number of sections increases, the rounded bumps along
its length and the slant of its width become less and less
distinct. The figure will approach the limit of being a rectangle.
3.5.1: Circumference and Area of a Circle7
Slide 9
Show how the perimeter of a hexagon can be used to find an
estimate for the circumference of a circle that has a radius of 5
meters. Compare the estimate with the circles perimeter found by
using the formula C = 2 r. 3.5.1: Circumference and Area of a
Circle8
Slide 10
Draw a circle and inscribe a regular hexagon in the circle.
Find the length of one side of the hexagon and multiply that length
by 6 to find the hexagons perimeter. 3.5.1: Circumference and Area
of a Circle9
Slide 11
Create a triangle with a vertex at the center of the circle.
Draw two line segments from the center of the circle to vertices
that are next to each other on the hexagon. 3.5.1: Circumference
and Area of a Circle10
Slide 12
To find the length of, first determine the known lengths of and
Both lengths are equal to the radius of circle P, 5 meters. 3.5.1:
Circumference and Area of a Circle11
Slide 13
Determine The hexagon has 6 sides. A central angle drawn from P
will be equal to one-sixth of the number of degrees in circle P.
The measure of is 60. 3.5.1: Circumference and Area of a
Circle12
Slide 14
Use trigonometry to find the length of Make a right triangle
inside of by drawing a perpendicular line, or altitude, from P to.
3.5.1: Circumference and Area of a Circle13
Slide 15
Determine bisects, or cuts in half,. Since the measure of was
found to be 60, divide 60 by 2 to determine The measure of is 30.
3.5.1: Circumference and Area of a Circle14
Slide 16
Use trigonometry to find the length of and multiply that value
by 2 to find the length of is opposite. The length of the
hypotenuse,, is 5 meters. The trigonometry ratio that uses the
opposite and hypotenuse lengths is sine. 3.5.1: Circumference and
Area of a Circle15
Slide 17
The length of is 2.5 meters. 3.5.1: Circumference and Area of a
Circle16 Substitute the sine of 30. Multiply both sides of the
equation by 5.
Slide 18
Since is twice the length of, multiply 2.5 by 2. The length of
is 5 meters. 3.5.1: Circumference and Area of a Circle17
Slide 19
Find the perimeter of the hexagon. The perimeter of the hexagon
is 30 meters. 3.5.1: Circumference and Area of a Circle18
Slide 20
Compare the estimate with the calculated circumference of the
circle. Calculate the circumference. 3.5.1: Circumference and Area
of a Circle19 Formula for circumference Substitute 5 for r.
Slide 21
Find the difference between the perimeter of the hexagon and
the circumference of the circle. The formula for circumference
gives a calculation that is 1.416 meters longer than the perimeter
of the hexagon. You can show this as a percentage difference
between the two values. 3.5.1: Circumference and Area of a
Circle20
Slide 22
From a proportional perspective, the circumference calculation
is approximately 4.51% larger than the estimate that came from
using the perimeter of the hexagon. If you inscribed a regular
polygon with more side lengths than a hexagon, the perimeter of the
polygon would be closer in value to the circumference of the
circle. 3.5.1: Circumference and Area of a Circle21
Slide 23
Show how the area of a hexagon can be used to find an estimate
for the area of a circle that has a radius of 5 meters. Compare the
estimate with the circles area found by using the formula 3.5.1:
Circumference and Area of a Circle22