Circumference of a Circles

REVIEW

NAME MY PARTS

Tangent

– Line which intersects the circle at exactly one point.

Point of Tangency

– the point where the tangent line and the circle intersect (C)

C

DL

M

Secant – Line which intersects the circle at exactly two points.e.g. DL

NAME EACH OF THE FOLLOWING:

EA

BD

C

O

1. A Circle

AnswerCircle O

NAME EACH OF THE FOLLOWING:

EA

BD

C

O

2. All radii

AnswerAO, BO, CO

DO, EO

NAME EACH OF THE FOLLOWING:

EA

BD

C

O

3. All Diameters

AnswerAD and BE,

NAME EACH OF THE FOLLOWING:

EA

BD

C

O

4. A secant

AnswerBC

k

NAME EACH OF THE FOLLOWING:

EA

BD

C

O

5. A Tangent

AnswerEK

k

NAME EACH OF THE FOLLOWING:

EA

BD

C

O

5. Point of Tangency

AnswerE

k

CIRCUMFERENCECircumference – is a distance around a circle.

Circumference of a Circle is determined by the length of a radius and the value of pi.

The formula is

C= 2r or C = d

r

P

EXAMPLE 1

WHAT IS THE CIRCUMFERENCE OF A CIRCLE IF RADIUS

IS 11 cm?

11 cm

R

Solution:C = 2r

C = 2( 11 cm)C = 22cm orC = 69.08 cm

EXAMPLE 2

THE CIRCUMFERENCE OF A CIRCLE IS 14cm. HOW

LONG IS THE RADIUS?

r=?

R

Solution:C = 2r

14cm = 2rDividing both sides by 2 .7 cm = r orr = 7 cm

AREA of a Circles

INVESTIGATION

IS IT POSSIBLE TO COMPLETELY FILLED THE CIRCLE WITH A SQUARE REGIONS?

R

NO.

INVESTIGATION

HOW IS THE AREA OF THE CIRCLE MEASURED?

R

In terms of its RADIUS.

INVESTIGATIONTAKE A CIRCULAR PIECE OF PAPER CUT INTO 16 EQUAL PIECES AND

REARRANGE THESE PIECES

WHAT IS THE NEW FIGURE

FORMED?r

NOTICE THAT THE NEW FIGURE FORMED

RESEMBLES A PARALLELOGRAM.

• 1

• 2

• 5 • 7 • 9• 11 • 13• 3

•8 • 10 • 12 • 14• 4 • 6

The BASE is approximately equal to half the circumference of the

circular region.

base = C orb= r

2

1

r h= r

Area of 14 pieces = area of the //gram = bh

= r( r) = r²

• 1

• 2

• 5 • 7 • 9• 11 • 13• 3

•8 • 10 • 12 • 14• 4 • 6

base = C orb= r

2

1

r h= r

EXAMPLE 1

WHAT IS THE AREA OF A CIRCLE IF radius

IS 11 cm?

11 cm

R

Solution:A = r ² = ( 11 cm)²A = 121cm² or = 379.94 cm²

EXAMPLE 2

WHAT IS THE AREA OF A CIRCLE IF radius

IS 4 cm?

4 cm

R

Solution:A = r ² = ( 4 cm)²A = 16cm² or = 50.24 cm²

EXAMPLE 3THE CIRCUMFERENCE OF A CIRCLE IS

14cm. WHAT IS THE AREA OF THE CIRCLE?

Solution:Step 1. find r.C = 2r

14cm = 2rDividing both sides by 2 .7 cm = r orr = 7 cm

Step 2. find the areaA = r ² = ( 7 cm)²A = 49cm² or = 153.86 cm²

EXAMPLE 4THE CIRCUMFERENCE OF A CIRCLE IS

10cm. WHAT IS THE AREA OF THE CIRCLE?

Solution:Step 1. find r.C = 2r

10cm = 2rDividing both sides by 2 .5 cm = r orr = 5 cm

Step 2. find the areaA = r ² = ( 5 cm)²A = 25cm² or = 78.5 cm²

TRUE OR FALSE

• 1. All radii of a circle are congruent.

• ANSWER TRUE

TRUE OR FALSE

• 2. All radii have the same measure.

• ANSWER FALSE

TRUE OR FALSE

• 3. A secant contains a chord.

• ANSWER TRUE

TRUE OR FALSE

• 4. A chord is not a diameter.

• ANSWER TRUE

TRUE OR FALSE

• 5. A diameter is a chord.

• ANSWER TRUE

AREAS OF REGULAR

POLYGONS

REGULAR POLYGONS3 SIDES 4 SIDES 5 SIDES 6 SIDES

7 SIDES 8 SIDES 9 SIDES 10 SIDES

Given any circle, you can inscribed in it a regular polygon

of any number of sides.

It is also true that if you are given any regular polygon, you can circumscribe a circle about it.

This relationship between circles and regular polygons leads us to

the following definitions.

The center of a regular polygon is the center of the circumscribed

circle.

The radius of a regular polygon is the distance from the center to

the vertex.

The central angle of a regular polygon is an angle formed by

two radii.

12

The apothem of a regular polygon is the (perpendicular) distance from the center of the

polygon to a side.

APOTHEM( a)

NAME THE PARTS

12

THE CENTERTHE RADIUSCENTRAL

ANGLE

ANGLE 1 AND ANGLE 2

NAME THE PARTS

APOTHEM

AREAS OF REGULAR POLYGONS

The area of a regular polygon is equal to HALF the product of the APOTHEM and the PERIMETER.

A = ½ap where, a is the apothem and p is the

perimeter of a regular polygon.

FIND THE AREA OF A REGULAR HEXAGONS WITH A 9 cm

APOTHEM.

HINT:A radius of a

regular hexagon bisects the

vertex angle.

9 CM

REMEMBER:Each vertex angle regular hexagon is

equal to 120°. each vertex = S ÷ n

FIND THE AREA OF A REGULAR HEXAGONS WITH A 9 cm

APOTHEM.

3

9

SOLUTION:Use 30-60-90 ∆

½s = = 3

Multiply both sides by 2

S= 6

9 CM

60

½ s

3

3 So, perimeter is equals to 36 3

FIND THE AREA OF A REGULAR HEXAGONS WITH A 9 cm

APOTHEM.

SOLUTION:A = ½ap

= ½( 9cm)36 cm = ½( 324 cm² )

= 162 cm²

9 CM

60

½ s

33

So, perimeter is equals to 36 3

3

FIND THE AREA OF A REGULAR triangle with radius 4

4 CM60

½ s

3