spheres

Surface area of Spheres
Finding the surface area of a sphere
In lesson 10.7 a circle was described as a locus of points in a plane that are a given distance from a point. A sphere is the locus of points space that are a given distance from a point.
Finding the surface area of a sphere
A diameter is a chord that contains the center. As with all circles the terms radius and diameter also represent distances and the diameter is twice the radius
Theorem 12.11: Surface Area of s sphere
The surface area of a sphere with readius r is S=4πr2

Secants and tangents

Lesson 8-5
Central Angle
Central Angle Theorem
The measure of a center angle is equal to the measure of the intercepted arc.
Inscribed Angle
Intercepted Arc
Secants, Tangents, and Angle Measures
Secant – A line that intersects a circle in two points
Interior Angle Theorem
Exterior Angles
Exterior Angle Theorem
Example: Exterior Angle Theorem

May 16thh

•May 16th
•Finding the surface area of a pyramid.
•A pyramid is a polyhedron in which the base is a polygon and the lateral faces are triangles with a common vertex. The intersection of two lateral faces is a lateral edge. The intersection of the base and a lateral face is a base edge. The altitude or height of a pyramid is the perpendicular distance between the base and the vertex.
•More on Pyramids
A regular pyramid has a regular polygon for a base and its height meet the base at its center. The slant height of a regular pyramid is the altitude of any lateral face. A non-regular pyramid does not have a slant height.
•Ex 1. Finding the Area of Lateral Face.
•Architecture. The lateral faces of the Pyramid Arena in Memphis, Tennessee are covered with steel panels. Use the diagram of the arena to find the area of each lateral face of this regular pyramid.

Equations of a Circle

Objectives
Develop and use the equation of a circle.
Adjust the equation for a circle to move the center in a coordinate plane.
The Equation of a Circle
Given the definition of a circle: The set of all points in a plane that are equidistant from a given point called the center, we know that we are dealing with points in the coordinate plane that are a given distance, r, from a single point, the center, that we can represent with a set of coordinates that we’ll call (h, k) so that we can use them anywhere in the plane.
The Equation of a Circle, Cont.
Since we are working with Distance, the Distance Formula will be used.
The Equation of a Circle, Cont.
An equation for a circle with center at (h, k), and a radius of r units is:
Key Skills
Key Skills

May 5th

•Math Notes
•May 5th, 2008
•Area Investigation
•what is the area of a triangle?
•1/2 bh.
•What if the triangle is an equilateral triangle then what is the formula
•Area Of an Equilateral Triangle
•The area of an equalateral is one fourth the square of the length of the side times the square root of 3.
•Area Investagation
•In a regular polygon, how many triangles can be formed with a vertex at the center and one side along the side of the polygon?
•Definitions
•Apothem- the distance from the center to any side of the polygon.
•Area Investigation
•calculate the area of a regular pentagon, in terms of the side length, s, and the apothem.
•
•in terms of s and a what is the area of a heptagon?
•
•in terms of s and a, what is th earea of an ngon?
•Tetrahedron
•A regular four sided 3-D shape.
•4 faces
•6 vertices
•Tetrahedron net
•Tetrahedron model
•Spin a tetrahedron
•Area Investigation
•Calculate the area of a regular pentagon, in terms of the side, s, and the apothem
•
•The area of each triangle is 1/2 sa
•
•There are 5 triangles, so the total area is 1/2 sa (5)
•
•In terms of s and a what is the area of a heptagon?
•
•In terms of s and a what is the area of a n-gon?
•Cube
•6 faces
•8 vertices
•12 edges
•Cube net
•Cube model
•Spin a cube
•Octahedron
•8 faces
•6 vertices
•12 edges
•Octahedron net
•Octahedron model
•Spin a octahedron

12-1 & 12-2

•12-1 & 12-2: Space Figures, Nets, & Diagrams.
•Objective: To recognize nets and to make isometric drawings of space figures.
•Polyhedron
•A three dimensional figure whose surfaces are polygons.
- Faces: the polygons making up the sides.
-Edges: A segment that is formed by the intersection of two faces.
-Vertex: a point where the three or more edges intersect.
•Euler’s Formula
F+V=E+2
5+6=9+2
11=11
•Space figures and Nets
Net: A two dimensional pattern that you can fold to form a three dimensional figure.
•Cross-Sections
•Cross Section: the intersection of a solid and a plane.

notes for the 12th

12.2 Surface Area of Prisms & Cylinders
Prism
A polyhedron with two congruent faces (bases) that lie in parallel planes.
- The lateral faces are all parallelogram.
- The height is the perpendicular distance between the bases.
Surface Area: The sum of the area of its faces
Lateral Area: The sum of the area of the lateral faces.
Surface Area of a Right Prism
The surface area, S, of a right prism is:
S=2B+Ph
B= area of one base.
P= perimeter of one base.
H=height of the prism.
2B represents the “base area”
Ph represents the “lateral area”
Find lateral area & surface area

page 640 9-22

HOMEWORK

notes for the 5th

Area Investigation:
what is the area of a triangle?
1/2 bh.
What if the triangle is an equalateral triangle then what is the formula.

Area of an Equilateral Triangle:
The area of an equalateral is one fourth the square of the length of the side times the square root
of 3.

Area Investagation:
In a regular polygon, how many triangles can be formed with a vertex at the center and one side along the side of the polygon?

DEFENITIONS:
Apothem- the distance from the center to any side of the polygon.

Area investagation
* calculate the area of a regular pentagon, in terms of the side length, s, and the apothem.

* in terms of s and a what is the area of a heptagon?

* In terms of s and a, what is th earea of an ngon?

Area Investagation:
Calculate the area of a regular pentagon, in terms of the side, s, and the apothem

The area of each triangle is 1/2 sa

There are 5 triangles, so the total area is 1/2 sa (5)

In terms of s and a what is the area of a heptagon?

In terms of s and a what is the area of a n-gon?

BONUS

what are platonic solids and what did each represent in greek history?

NOTES, for the 2nd.

Vocabulary:
Face
Edge
Vertex
Base
Polyhedron
prism
pyramid
lateral surface
cylinder
cone
sphere
hemisphere

Three Dimensional Figures:
Three Dimensional Figures, or solids, have length, width, and height. A flat surface of a solid is a face. An edge is where two faces meet, and a vertex is where three or more edges meet. The face that is used to classify a solid is a base.

The surfaces of a three dimensional figure determine the type of solid it is. A polyhedron is a three deminsional figure whose surfaces, or faces, are all polygons. Prisms and pyramids are two types of polyhedrons.

Prism:
A prism is a polyhedron that has two parallel congruent bases. The bases can be any polygon. The other faces are parallelogram.

Pyramids:
A pyramid is a polyhedron that has one base. The base can be any polygon. The other faces are triangles.

Equation of a circle.

•Objectives
•Develop and use the equation of a circle.
•Adjust the equation for a circle to move the center in a coordinate plane.
•The Equation of a Circle
•Given the definition of a circle: The set of all points in a plane that are equidistant from a given point called the center, we know that we are dealing with points in the coordinate plane that are a given distance, r, from a single point, the center, that we can represent with a set of coordinates that we’ll call (h, k) so that we can use them anywhere in the plane.
•The Equation of a Circle, Cont.
•Since we are working with Distance, the Distance Formula will be used.
•The Equation of a Circle, Cont.
•An equation for a circle with center at (h, k), and a radius of r units is:

Cirlce in coordinate plane.

Objectives:

• Develop and use the equation of a circle.
• Adjust the equation of the circle to move the center ub a coordinate plane.

The Equation of a Circle:

• Given the definition of a circle: The set of all points in a plane that are equidistant from a given point called the center, we know that we are dealling with points in the coordinate plane that are a given distance, r, from a single point, the center, that we'll call (h,k) so that we can use them anywhere in the plane.
• since we are working with distance the distance formula will be used.
• Replace the letter with the new information we know.
• r= the square root of x-h squared + y - k squared.
• replaceing distance with the word radius.
• Square both sides and you'll get the radius squared=x-h squared + y-k squared
• an equation for a circle at (h,k) and a radius of r units is
• (x-h)squared+(y-k)squared=rsquared.

Key Skills:

Sketch a cirlce from its equation.

(x-2)2+(y-1)2=9

HOMEWORK, TEST ON FRIDAY

572-573
8-28 not 21

Segments of Tangents, Secants and Chords

External Secant Segment: That part of a secant segment exterior to the circle

Secant Segment: That part of a secant thta extends from an external point that includes a chord of the cirlce.

Tangent Segment: That part of a tamgent that extends from an exterior point to the point of tangency.

Theorem 10-15: If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

Theorem 9.5.2: If two secants intersect outside a circle. the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (Whole x Outside =Whole x Outside)

April 16th, 2008

Secant- Al line that intersects a circle in two points.

Secants create chords.

Interior Angle Theorem:
Definition: Angles that are formed by two intersecting chords.


The measure of the angle formed by two intersecting chords is equal to 1/2 the sum of the measure of the intercepted arcs.

Exterior Angles
An Angle formed by two secants and two tangents or a secant and a tangent drawn from a point outside the circle.

the measure of the angle formed is one 1/2 the distance of the intercepted arcs.

Homeworkkkk

556-557 8-26 NOT 21,22

Notes for the 15th

tangent to a circle- a tangent us a line that touches a circle in one spot.
point of tangency- the single point where that line touches the circle.


Tangent Theorem 10-9: If a line is tangent to a circle then the line is perpendicular to a radius of the circle drawn to the point of tangency.

Theorem 10-10: If a line is perpendicular to a radius of a circle at its endpoint on the circle, then hte line is a tangent to the circle

Theorem 10-11: If two segments from the same exterior point are tangent to a circle then they are congruent.

STUDYYY FOR THE TEST.

you neeed to study for the testttt.

Homework for the 14th

13) 32 degrees
37 degrees

14) 56 d
78 d
32 d
41 d
14 d
25 d
36 d
89 d

GEOMETRY NOTES 4/14

inscribed angle- an inscribed angle is an angle made up of two chords. Since it is made up of chords. The vertex of the angle will be ON the circle.
There are three types of inscribed angles center interior, center exterior, Center included.
Intercepted Arc- the mino acr defined by the two endpoints of chords forming an inscribed angle that are not part of the vertex of the inscribed angle.
Inscribed angle therorem 10-5: The measure if n angle inscribed in a circle is equal to half the measure of an intercepted arc (or, the measure of an intercepted arc is twice the measure of the inscribed angle)
Theorem 10-6: Arc Inctercept Theroem: If two inscribed angles intercept congruent arcs of the same arc the angles are congruent.
Theorem 10-7: Right-Angle: If an inscribed angle intercepts a semicircle. then the angle is a right angle.

PAGE 526 21-57

21 TUVWXZ

22 ZV

23 ZW

24 TX

25 ZV

26 4ft?

27 2.5 ?

28. 60cm.

29. 16in

30. 9

31. .06

32. 6

33. 8

34 6

37. 35

38. 8

39. 7

40. 15

41. 26

42. 18

41. 6

42. 12

43. 16

44. 5

45.6

46. 15

47. 19

48 16

49. 13

50. 18

51. 52

53. 14

54. 41

Homework for april 1st

pp 450-451
13-36

NOTES FOR APRIL 1ST, OH8

Position of The figure

-Position the figure to simplify the proof

Find the missing coordinate
-Opposite sides of a parallelogram are parallel & congruent, therefore the y coordinate od D must be C.
-The length of AB is a, and the length of DC is a, so rhe x coordinate of D must be (a+b)-a or b
-Thus the coordinate of D are (b,c)

If you are working with a parallelogram and you say each side top and bottom is a units long its easy for you to find this point, i'd say that it is C units long if the lower corner is a it mus be (a,0) because you didn't move around the distance. The parrallelogram is C units tall and you wanna find the top corner coordinate ( , ) so lower left hand corner is b so the coordinate corner of the dotted line is (b,c) so top corn is now ( ,c) how far did you move to get from there to there? b units so it is now (b,c) so the entire distance is (a+b,0) the one above is (a+b,c)

Coordinate Proof
Given a square ABCD with midpoints MNPQ
Proove MNPQ is a square.

How do we proove this?
4 congruent sides
4 congruent angles
prove its a rhombus
prove its a rectangle
then if it has this it is a square.

Prove that the diagnols are congruent.
Prove the diagnols are perpendicular.

Extra Credit Assignment

Archimedes discovereed a method for estimating the value for pie. How did he do this? Givee a detailed explanation in your own words. Must be one page long excluding any drawings with sources listed on a seperate sheet DUE 4/4/08!!

407-408,, 13-41

13. 40

14. 20

15. 45

16.28

17. 85

18. 66

19. 24

20. 55

21. 59

22. 58

23. 24

24. 75

27. The measure of angle m= 65, r=75, p=55, q=41

28.The measure of angle e=87, f=56, g=41, h=25. j=63

29. The measure of angle m= 65, r=75, p=55, q=41

30. The measure of angle m= 65, r=75, p=55, q=41

31. The measure of angle m= 65, r=75, p=55, q=41

Homework for the

Pages 421-423 10-36

10) Yes it is.

11) Proof, this is a parallelogram because i said so.

12) i don't know

13) yes this is a parallelogram

14) no this is not a parallelogram

15) no this is not a parallelogram

16) yes this is a parallelogram

17) no this is a parallelogram

18) yes this is a parallelogram

19) x=10 y=6

20) x=7 y=5

21) x=8 y=4

22) x=14 y=7

23) x=12 y=6

24) x=2 y=11

25) yes

26) no

27) yes

28) yes

29) no

30) no

31) yes

32) no

33)idkk

34)idkk

Notes: March 27th

Theorem 8-20: The Median of a trapezoid is parallel to the bases and its measure is one-half the sum of the measures of the bases

Homework for the 27thh

9) yes, no

10) yes, yes

11) no, no

12) yes, yes

13) DE= 13

14) VT= 9

15)W= 80
7= 100

16) Q= 60
S= 95

i don't understand 17-25! :(

Homework for March 27th

PAGES: 442-443 9-25

GEOMETRY HOMEWORK FOR THE 26TH.

PAGES 434-435 NUMBERS 12-34

12) 60 degrees

13) 30 degrees

14) 45 degrees

15) 45 degrees

16) 60 degrees

17) 30 degrees

18) 45 degrees

19) 30 degrees

20)rhombus

21)rectangle

22)square

23)rectangle

24 & 25 CAN'T DO.

26) always

27) sometimes

28) never

29) sometimes

30)sometimes

31) never

32- 34 CAN'T DO.

35) Theorem 8-16: If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus.

THEN THE PARALLELOGRAM WOULD BE SLANTED AND THE ANGLES WOULD BE EQUAL AND IT WOULD BE A RHOMBUS.

online geom

ONLINE GEOMETRY BOOK AND PASSWORD.

GEOM05
N6despez

Notes For March 26th

Theorem 8-17: If the diagonals of a parallelogram bisect the opposite angles of the parallelogram then the parallelogram is a rhombus

Summary Rhombi

1) A rhombus has all the properties of a parallelogram.

2) All sides are congruent

3) Diagonals are perpendicular

4)Diagonals bisect the angles of the rhombus

Summary Squares

1) A square has all the properties of a parallelogram

2) A square has all of the properties of the rectangle

3) A square has all of the properties of the rhombus




Trapezoid

What is a trapezoid?
A trapezoid is a quadrilateral with exactly one set of parallel sides

Trapezoid Bases: The parallel sides of the trapezoid.

Trapezoid Legs: The non-parallel sides of a trapezoid

Isosceles Trapezoid: If the legz of a trapezoid are congruent then the trapezoid is isoceles.

Median of a trapezoid- Is the segment that connects the medpoints of the legs of a trapezoid.

Theorems, Postulates, & Definitions

• Theorem 8-18: Both pairs of base angles of an isosceles trapezoid are congruent
• Theorem 8-19: The diagonals of an isosceles trapezoid are congruent.

Homework For March 26th

pages434-435
12-34

March 6th

Math Notes March 6, 2008

Polygon Angle Sum Theorems
Objectives:
· To Classify Polygons
· To find the sum of the measures of the interior and exterior angles.
Polygon:
· A closed plane figure.
· With at least 3 sides (segments)
· The sides only intersect at their endpoints
· Name it by starting at a vertex & go around the figure clockwise or counterclockwise listing each vertex you come across.
Example one:
II. Also Classify polygons by their shape
a) Convex Polygon: Has no diagonal
With points outside the polygon.
b) Concave Polygon: Has at least
One diagonal with points outside the polygon.

a)

March 11th

Math Notes March 11, 2007
Using a calculator will give you an approximation.

March 17th

· Math Notes;
· Check to see if it is a rectangle as well as a rhombus:
· W(1,10); X(-4,0); Y(7,2); Z(12,12)
· The diagonals are not congruent, so this is not a rectangle or a square.

March 17th

· Math Notes;
· Check to see if it is a rectangle as well as a rhombus:
· W(1,10); X(-4,0); Y(7,2); Z(12,12)
· The diagonals are not congruent, so this is not a rectangle or a square.

Geom Notes For SQUARE ROOTS! ;]

Roots
Square Roots
When working a square root problem. Ask: -- what times itself is the number inside the root symbol?”
=3 because 3 times 3 is 9
because 5 times 5 is 25
Roots and Prime Numbers
=3 because 3x3x3 is 27. The small 3 outside the root symbol tells how many times the answer must be multiplied to get the number inside the root.
because 2x2x2x2x2 or 25=32
Prime Number: a number with only factor one and itself 2,3,5,7,9,11,13,1,7,19,23… are prime numbers fifteen is not prime because 3 and 5 also divide it evenly. 15 is a composite number.
Prime factorization.
Prime Factorization is writing a number using multiplication of only prime numbers.
12 can be written as 3x4 but 4 is not prime and can be written as 2x2
So the prime factorization of 12 is 3x2x2 this can also be written 3x22
Graphing Prime Numbers

330

10 x 33
5 x2 3 x 11
To write 330 using its prime factorization start breaking it down by each number.
Simplifying Roots
You won’t be using the radical button on your calculator anymore.
= 2
Prime Factorization

Notes for March 25th, 2008

Definition- Rectangle
A rectangle is a parallelogram with four right angles.

PROPERTIES
1. Opposite sides are congruent and parallel.
2. Opposite angles are congruent.
3. Consecutive angles are supplementary.
4. Diagonals bisect each other and are congruent.
5. All four angles are right angles.

Theorems, Postulates, & definitions.

Theorem 8-13: If a parallelogram is a rectangles the diagonals are congruent.
Note: If one angle of a parallelogram is a right angle then the parallelogram is a rectangle.
The Housebuilder Theorem 8-14: If the diagonals of a parallelogram are congruent then the parallelogram are congruent then the parallelogram is a rectangle.
Rhombi and Squares
Theorem 8-15: The diagonals of a rhombus is perpendicular.
Theorem 8-16: If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus.

Homework for March 19th.

page 427 numbers 10-35

Geometry Notes, March 19th

Theorems, Postulates, and Definitions.


Theorem 8-8: If two pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 8-10: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 8-11: If the diagonals of a quadrilateral bisect each other then the quadrilateral is a parallelogram.


Theorem 8-12: If one pair of opposite sides of a quadrilateral are parellel and congruent, then the quadrilateral is a parrelellagram.

Summary!

A quadrilateral is a parallelogram if any one of the following statements are true.
1. Both pairrs of opposite sides are parrallel. (definition)
2. Both pairs of opposite sides are congruent. (theorem 8-9)
3. Both pairs of opposite angles are congruent (throrem -10)
4. The diagnols bisect each other (theorem 8-11)
5. A pair of opposite sides is both parellel and congruent (theorem 8-12)

Parallelograms in the coordinate plane

• The slope formula can be used to determine if the opposite sides have the same slope.
• The distance formula can be used to see if the opposite sides are congruent or
• The slope and the distance formula ccan be used to determine if one pair of opposite sides is parallel and congruent.