Aim: How do you find the surface area and volume of a sphere?
April 18, 2012


Spheres: A Sphere is the set of all points in space equidistant from a given point called the center.

Volume:                                                                                       Surface Area:

                                      



Surface Area:

Aim: How do we find the volume of pyramids and cones?
April 17, 2012

Volume of a Pyramid

A pyramid has a base and triangular sides which rise to meet at the same point. The base may be any polygon such as a square, rectangle, triangle, etc. The general formula for the volume of a pyramid is:

Area of the base * Height * 1/3

The volume of a pyramid with a rectangular base is equal to: 

Length_of_base * Width_of_base * Height * 1/3

                                           

Volume of a Cone

The volume of a cone is 1/3(Area of Base)(height) = 1/3 π r² h

                                          

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

A faster way to find lateral area:

L.A = ph

Surface area = lateral area + area of the bases
 ( S.A = L.A + 2B)

The surface area of a prism = 2 × area of base  +  perimeter of base × H

The actual formula used to find the surface area will depend on the shape of the base of the prism.

Aim: How do we identify polygons?
April 02, 2012




 Solid Geometry
Solid geometry is the geometry of the 3-dimensional space.
It is called three-dimensional, or 3D because there are three dimensions width, depth and height.

Properties:
Solids have properties, such as
- volume
- surface area

There are two main types of solids, "polyhedra" and "non-polyhedra"

Polyhedra- They must have have flat faces
 - prisms , pyramids , and platonic solids

Non-polyhedra- if any surface is not flat.
- sphere , cylinder and cone

Prisms: A prism has the same cross section all along its path/length.

Cross sections : A cross section is the shape you get when cutting straight across an object.

A prism is a solid with bases that are 2 congruent polygons. The other sides of the prism are called the lateral faces.

A prism is named by the shape of its base.

Aim: How do we find the area of a circle?

Circle area conjecture: The area of a circle is given by the formula, Area = πr² , where A is the area, r is the radius of the circle.

       Example:

 

Aim:How do we find the area of regular polygons?
March 19, 2012


The area of a regular polygon:
   You can divide a regular polygon into congruent isosceles triangles by drawing segments to each vertex.

Base: S (side)        Altitude: A (apothem)

The variable A appears in the formula for regular polygons. It stands for the length called the apothem, the perpendicular segment from the center to the side of the polygon.


Regular polygon area conjecture:
    The area of a regular polygon is given by the formula 1/2nas, where A is the area, a is the apothem, n is the number of sides ans S is the length.

                   formula: 1/2nas or p= 1/2Pa

Aim:How do we find the area of parallelograms, kites and trapezoids?

Area of parallelograms: 
       Area=base x height
                       

Area of trapezoids:
        Area=1/2 x height (b1 + b2)


                                      




















Area of kite: 2 pairs of consecutive congruent sides.
               Area= d1 + d2/2

                                        

Aim:How do we calculate the area of rectangles and triangles?
March 12, 2012


The area of a plane figure is the measure of the region enclosed by the figure. You measure the area of a figure by counting the number of tiles inside.

triangles:
triangle area conjecture - the area of  the triangle is given by the formula a= 1/2bh, where A is the area, B is the length of the base and H is the height of the triangle.

rectangles:

How to find the area of a rectangle:

• The area of a rectangle can be found by multiplying the base times the height.
• If a rectangle has a base of length 6 inches and a height of 4 inches, its area is 6*4=24 square inches

Aim: How do we find compound loci?
March 07, 2012

 compound locus- problem involves two, or possibly more, locus conditions occurring at the same time.  The different conditions in a compound locus problem are generally separated by the word "AND" or the words "AND ALSO".




5 rules of locus:

• 1 point - circle 
• 2 points - 1 line
• 1 line - 2 lines
• 2 lines - 1 line
• two intersecting lines - two intersecting lines 

Aim: How do we find the locus of points?


The locus is the set of all points that satisfy a given condition. A locus is a general graph of a given equation.

The locus of points equidistant from a single point is a set of points, equidistant from the point in every direction

The locus of points equidistant from two points is the perpendicular bisector of the line segment connecting the two points

The locus of points equidistant from a line are two lines, on opposite sides, equidistant and parallel to that line

The locus of points equidistant from two parallel lines is another line, half-way between both lines, and parallel to each of them

                       The locus of points at a fixed distance, d, from point P is a circle with the given point P as its center and d as its radius.

Aim: What is a mathematical statement?


A mathematical statement is a statement that can be judged true or false.
and:both statements must be true for the statement to be true completely.

There are 4 conditionals:
1.the conditional
2.the inverse
3.the converse
4.contrapositive

Aim:What is logic?


LOGIC is THINKING!!!


Logic is the tool to determine between true or false.
*Inductive reasoning- when your not sure what comes after.

You use logic everyday to...
solve geometry problems
crossing the street 
and moreee!

Aim:How do we use other definitions transformations?


Glide Reflection: The combination of a reflection in a line and a translation along that line.




Orientation: Refers to the arrangements of the points, relative to one another , after a transformation has occured.


Invariant: A figure or property that remains unchanged under a transformation of the plane is referred to as invariant. (something that stays the same and doesn't change)


An opposite isometry changes the order.




A direct isometry preserves orientation the letters go in the same clockwise and counterclockwise direction on the firgure and its image.

Aim:How do we solve composition of transformation problems?

Composition Of Transformations 

When two or more transformations are combined to form a new transformation, the result is called a composition of transformation.It follows a translation.

Ex: Find the coordinates of the image of (2,4) under the transformation ry-axis oT(3,-5)

(2,4)

(3,-5)   = (5,-1)

The symbol for a composition of transformations is an open circle. 

How do we graph dilations?

Aim:How do we graph dilations?


Dilations:A type of transformation that causes an image to stretch or shrink in proportion to its original size.

Scale factor: the ratio by which the image stratches or shrinks is known as the scale factor.

- Multiply the dimensions of the original image by the scale factor to get the dimensions of the dilated image.

Ex: Find the image of (3,-2) under the dilation2
               3x2=6       -2x2=-4
                       (6,-4)

How do we identify transformations?

How do we identify transformations?

 A transformation is when you move a geometric figure. There are 4 types of transformations:

1.Translation: Every point is moved the same distance in the same direction.
2.Reflection: Figure is flipped over a line of symmetry.
3.Rotation: Figure is turned around a single point.
4.Dilation: An enlargement or reduction in size of image.