5/3/12 How do we find the area of sectors and other parts of a circle?

How do we find the area of sectors and other parts of a circle?

Sector of a circle
- a sector of a circle is the region between the radii and the arc of a sector
To find the area of a sector : # of degrees /360 x (pi) r^2
Ex:

Segment of a circle 
- a segment of a circle is the region between a chord and arc of circle
To find the area of the segment of a circle : # of degrees/360 x  (pi)r^ - 1/2bh
Ex:

Annulus 
- an annulus is the region between two concentric circles
To find the area of the annulus : (pi)R^2- (pi)r1

Ex:

Practice question : If radius 1 is 12 and radius 2 is 6, What is the area of the annulus?

                                                           Citations
- Problem solving , http://www.p12.nysed.gov/ciai/mst/math/sampletasks/Math6Sample.htm
- Chord and segment of a circle, http://www.winpossible.com/lessons/Chord_and_Segment_Of_A_Circle.aspx, 2011
- Annulus calculator , http://www.calculatorsoup.com/calculators/geometry-plane/annulus.php

5/1/12 How do we use secant , chord and tangents in circles? ?

How do we use Secants, Chords and Tangents in circles?

Secant
- a line that cuts through or divides a curve into two or more parts
Ex:

Chord
- a straight line thats joins the ends of an arc
Ex :

Tangent
-  straight line or plane that touches a curve or curved surface at a point
Ex:

Practice Question : What does the line TL image below represent?

                                                                       Citations 
- Chord , Wolfram MathWorld , http://mathworld.wolfram.com/Chord.html , 2012

- http://www.google.com/imgres?q=secant&start=173&um=1&hl=en&sa=N&biw=1280&bih=861&addh=36&tbm=isch&tbnid=Lo2YH7az3PTd1M:&imgrefurl=https://internal.shenton.wa.edu.au/maths/WestOne3CMAS/content/004_functions/media/glossary.html&docid=r1W_zcdlQ_KVDM&imgurl=https://internal.shenton.wa.edu.au/maths/WestOne3CMAS/content/004_functions/media/images/glossary_secant.gif&w=290&h=250&ei=QcymT8myJqGJ6QGyiaCdBA&zoom=1&iact=hc&vpx=606&vpy=126&dur=282&hovh=179&hovw=216&tx=126&ty=147&sig=100793830460579110965&page=8&tbnh=152&tbnw=195&ndsp=25&ved=1t:429,r:7,s:173,i:194

- Chapter 9: Chords and Arcs, http://whites-geometry-wiki.wikispaces.com/PETO626
- Geometry : Circles , http://www.sparknotes.com/math/geometry1/circles/section3.rhtml , 2012

How do we review transformations?

Isometry

- length is preserved

- sides are congruents ( like their reflecting)

Ex:                         

              

Direct isometry 

- orientation is preserved( stays the same)

-  The order of the lettering in the figure and the images are the same 

- either clockwise becomes counter clockwise or counter clockwise becomes clockwise 

- every translation and rotation is a direct isometry

Ex: 

Opposite isometry

- orientation isn't preserved (changes)

-  the order of the lettering is reversed

- every reflection and glide reflection is an opposite isometry 

 Ex:

Same orientation :

 rorgin (x,y) -> (-x,-y)              

 R90 (x,y) -> (-y,x)

 R180 ( x,y)-> (-x,-y)

 R270 ( x,y) -> ( y,-x)

Reversed orientation:

 r- x-axis (x,y) -> ( x,-y)

 r-y-axis (x,y) ->(-x,y)

ry = x (x,y) -> ( y,x)

ry = -x (x,y) -> (-y,-x)

Practice question : What is the difference between a direct and opposite isometry?

                                                      Citations 

- http://zsolania.blogspot.com/

- Haskell , Kenn, Functional Lens , http://www.kennknowles.com/blog/2007/12/03/calculating-the-reflect-rotate-translate-normal-form-for-an-isometry-of-the-plane-in-haskell-and-verifying-it-with-quickcheck/

4/24/12 What are altitudes , perpendicular bisectors and angle bisectors?

What are Altitudes, Perpendicular bisectors and Angle bisectors?

                     


                                       Angle bisectors


- an angle bisector,bisects an angle in half creating two congruent angles
Ex:

                            Perpendicular Bisectors
- a line that cuts through another line going in the opposite direction or bisects a side
Ex:

                                           Altitudes 
- a line segment through a vertex that is perpendicular to a line containing side
- makes a right angle
- is the height of the triangle

Ex:

Practice question : If the given height of the triangle is 14, What is the length of the altitude ? Explain your answer.











                                                        Citations


       -Triangle angle  bisector theorem, Math warehouse ,http://www.mathwarehouse.com/geometry/similar/triangles/angle-bisector-theorem.php

- Perpendicular bisector - interactive applet , http://www.analyzemath.com/Geometry/PerpendicularBisector/PerpendicularBisector.html , April ,3 2011

- Triangles , bisectors and circumcircles - interactive applet , http://www.analyzemath.com/Geometry/Circumcircle/Circumcircle.html , April ,3 2011

- Miller , Maria , Altitude of a triangle , homeschoolmath.net , http://www.homeschoolmath.net/teaching/g/altitude.php , 2003-2012

4/18/12 How do you find the surface area and volume of a sphere?

How do you find the surface area and volume of a sphere?

                                            Spheres 
- a sphere is a set of all points in space equidistant from a point called the center
Ex :

- In order to calculate the surface area of a sphere use the formula : SA= 4(pi)r^2
      - 'r' represents the radius of the sphere
In order to calculate the volume of a sphere use the formula : 4(pi)r^2/3
      -'r' represents the radius

Practice question: If the given radius of a sphere is 4 , What is the surface area of the sphere?

                                               Citations
Unit math example: ( Geometric solids) , http://www.unitmath.com/um/p/Examples/GeometricSolids/GeometricSolids.html ,  9/10/2000

3/29/12 How do we find the the surface area and lateral area of pyramids and cones?

How do we find the surface area and lateral area of pyramids and cones?

 Pyramids
- a pyramid is a solid that connects a polygon base to a point
- the slant height of a pyramid is the height of the lateral faces
Ex :

- you can calulate the lateral area of a pyramid by using the formula LA= 1/2pl
-'p' represents the perimeter
-'l' represents the slant height
- You can calculate the surface area


Cones
- the base of a cone is circle and the other end ( vertex) is pointed similar to a pyramid
- the slant height of a cone is the distance from the vertex to a point on the base
Ex:

- to calculate the lateral area of a cone use the formula (pi)rl
- 'r' represents the radius
-'l' represents the slant height
- In order to calculate the surface area , and extra steps need to be included
- use the formula SA= LA(lateral area) + B (base)
- to find the base use (pi)r^2 , then plug in the final number in the formula above this one

Practice Question: If the radius of a cone 'r' is 4 and the slant height 'l' is 12 , What is the the lateral area of a cone?










                                                         Citations
- Geometry help!!! triangular pyramid , MHF,http://mathhelpforum.com/geometry/27562-solved-geometry-help-triangular-pyramid.html , 2005-2012

- Ring my bell- Volume of cone , Lesson poly , Silicon valley education foundation, http://www.lessonopoly.org/node/2299 , 2010

3/27/12 How do we find surface area and lateral area of prisms and cylinders?

How do we find surface area and lateral area of prisms and cylinders?

 Prisms
- a prism is a solid figure with bases that are 2 congruent polygons
- the other sides of the prism are called  lateral faces
- a prism  is named by the shape of its bases
Ex:

- calculate the surface area of a prism by using the formula SA= LA+ 2B
 - " LA" represents the lateral area
- " B' represents the base
- Calculate the lateral are of a prism by using LA = ph
 - 'P' represents perimeter of a base
 - 'H' represents the height

Cylinders
Ex:

- calculate the lateral area using the formula 2(pi)rh
-'r' represents the radius
-' h' represents the height

- In order to calculate the the surface area multiple steps are made
- calculate the area of the base by using (pi)r^2
         - 'r' represents the radius
- multiple your answer by 2
- you then take your lateral area and add that with the number of the bases
    - you can go by this formula to calculate the sureface area of a cylinder  SA= LA +2B
-'LA' represents the lateral area
-'B' represents the base

Practice question : If the radius 'r' of a cylinder is 6 and the height ' h' is 4 , what is the lateral area of a cylinder?



                                           




                                            Citations 
-The 48 special crystal forms , University of wisconsin , greenbay , 1/20/11 , http://www.uwgb.edu/dutchs/symmetry/xlforms.htm

- Volume of a cylinder , Math homework series on geometry help , http://www.geometry-help.info/Volume_of_a_Cylinder.html, 2009