(4-17-2012)Aim: How do we find the volume of cones?

(4-17-2012)Aim: How do we find the volume of cones?

Cone 
The volume of a cone is shown by the formula v=1/3πr²h

http://3dshapes.org/images/stories/3dshapesclipart/thumbs-medium/cone1.pngExample:
What is the volume of a cone with a radius of 3 and a height of 9?
Leave it in the terms of π.
Answer:
v=1/3πr²h
v=1/3π3²(9)
=π9(9)/3
=81π/3
=27π units^3

Try it yourself!
What is the volume of a cone with a radius of 2 and a height of 7?

(4-18-2012)Aim: How do you find the surface area and volume of a sphere?

(4-18-2012)Aim: How do you find the surface area and volume of a sphere?

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

Surface area and volume formula for a sphere:

http://0.tqn.com/d/math/1/0/I/F/spherer.gif
Examples:
Surface Area
What is the surface area of a sphere with radius 5?
 leave it in the terms of π.
SA: 4πr²
4π5²
4π25
100πunits²

Volume:
What is the volume of a sphere with a radius of 8ft?
4πr^3 /3
4π8^3/3
4π512/3
2048π/3
=682.67πft^3 or 2143.6 ft ^3

Try it yourself!
Find the volume and surface area of 7in?
















                                         

Aim:How do we calculate the surface areas of cylinders

Aim: How do we calculate the surface area of cylinders?

The surface(base) of a cylinder is a circle?

It's area is (pie)r(radius)^2

 Example: (pie)6(radius)^2=36(pie)

 

 

http://jwilson.coe.uga.edu/emt669/Student.Folders/Pearman.lori/essay1/Essay11.gif

The surface area formula of a cylinder is : L.A(lateral area)+2B(base area (pie)r^2)

 

 

http://www.fea-optimization.com/ETBX/solidprop_help_files/surfarea.gif

The lateral area formula of a cylinder is: L.A (pie)d(diameter)h or 2(pie)rh

Example Questions:

1. A cylinder has the height of 15ft and radius of 6 ft? What is the lateral area?

L.A=(pie)dh

(pie)12*15

= answer: (pie)180ft^2 or 565.2ft^2

*reminder: don't forget to add the units*

 2. A cylinder has the Lateral Area of 300in, an base area of 78.5in, a height of 12in and radius of 4? What is the surface area?

S.A.=L.A. +2B

=300+2(78.5)

=300+157

=457in^2

TRY IT YOURSELF!

A cylinder has the height of 14cm and the radius of 5cm? What is the surface area and lateral area of the cylinder?

from household products to geometry figures

Cylinder:

  A cylinder is a non-polyhedral or a three dimensional figure that doesn't have a surface that is flat due to the fact that it's surfaces are curved.  Both bases are parallel and congruent and are circles rather than polygons.

Formula's for a cylinder: 

Area of base: 2(pie)r(radius)^2, Lateral Area: 2(pie)r(radius)h(height),  Surface Area: L.A(lateral Area)+2B(area of base(s)),

 Trapezoidal prism

A trapezoidal prism is a six faced three dimensional prism that has six faces which consists of two trapezoids and four rectangles and is also a polyhedra or three dimensional figure with a flat face.

Formula's for a trapezoidal prism: 

Area of Base: B1+B2/2(h), Lateral Area: p(perimeter)h, Surface Area: L.A+2B.

Sphere

 

A sphere is defined as the set of all points in three-dimensional Euclidean space that are located at a distance (the "radius") from a given point (the"center"). Twice the radius is called the diameter, and pairs of points on the sphere on opposite sides of a diameter are called antipodes . A sphere is also a non-polyherdra as well. 

Sphere's formula's:

Surface Area

, formula for the radius:

Square prism

A square prism is a three dimensional polyhedra with six congruent faces, faces that join along their edges or line segment were two edges meet and vertex's meet as well or point where three edges meet, a square prism has eight vertices.

Square prism's formula's:

Area of Base: s^2,Lateral Area: ph, Surface Area: L.A+2B

Rectangular Prism

A rectangular prism is a six sided three dimensional polyhedra a figure that has six rectangles, this is known as a prism because it has the same cross section along this figure.

Rectangular prism's formula's:

Area of base: B(base)*h, Lateral Area: ph, Surface Area: L.A. +2B or 2lw+2lh+2hw(l:length.w:width,and h:height)

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

(3-20-2012)Aim: How do we find the area of a circle?

Circle Area Conjecture
   The area of a circle is given by the formula A= πr^2, where A is the area and r  is the radius of the circle.
  The circumference of the circle is given by the formula C=πd.  Where c is the circumference and d is the diameter of the circle.

Area Example

A= πr^2
      π4^2(4*4)
      Answer:16πcm^2
          or
   Answer:16π= 50.24cm^2












Circumference example

A chocolate chip cookie has a diameter of 3in and a butterscotch cookie has a diameter of 5in. How much larger in circumference of the butterscotch cookie is to the chocolate chip cookie?

                         C= πdChocolate chip: C=3π in
 Butterscotch:    C=5π in
                         5π -3π= 2π in^2
                         Answer: 2π in^2 
                             or
Chocolatechip: 3π = 9.42 in
Butterscotch:   5 π = 15.7 in
                       15.7-9.42= 6.28
                        Answer: 6.28 in^2
              Answers: 2π in^2 or 6.28 in^2

TRY IT YOURSELF:
          A circle has the diameter of 14cm. What is the circumference and area of the circle?
*Remember the radius is half the diameter.   

Answer: 

Circumference: 14π 

Area: 196π

Aim: How do we find the area of regular polygons?

(3-19-2012)Aim: How do we find the area of regular polygons?

Area of an isosceles triangle with altitude a, and base s.

Area of a regular polygon
You can divide a regular polygon into congruent isosceles triangle by drawing segments in a vertex.

https://www.cdli.ca/courses/math1204/unit06_org02_ilo02/les02_002.gif 




Area of each isoscles triangle with altitude regular pentagon
A= sa/2(?)-number or amount of sides the figure has or 1/2as(?)-number or amount of sides the figure has 
a--altitude
s--side
A--area

Perimeter: (?)s
(?)--number of sides
s--side value

http://doversherborn.comcastbiz.net/highschool/academics/math/baroody/GeometryHonors/Class%20Notes/Chapter%2011/Lesson11-5/RegularPentagon.gif









For pentagon: A=sa/2(5)--Because there are five sides of the pentagon
Perimeter formula for pentagon: 5s



Area of hexagon
as/2(6) or 1/2as(6)
Perimeter:6s

http://jwilson.coe.uga.edu/emt668/EMAT6680.2003.fall/Montgomery/EMAT6690/Instructional%20Unit/Area/InstructUnitArea_files/image217.gif





Regular Polygon Area Conjecture
           The area of a polygon is given by the formula 1/2P*a, where a is the area, p is the perimeter. Also the formula that is more frequently used  1/2nas or nas/2 in which a is the apothem. The apothem is a perpendicular segment from the center to the side of the polygon.
Example:

 




http://www.capitan.k12.nm.us/teachers/shearerk/images/Apothem-hexagon.gif


S stands for the length of each side and n  is the number of sides of the polygon. 

Eamples:
1.
A=?
s=25in
a=25.8in
n=8in

A=nas/2
(8*25.8*25)/2
(5160)/2
Answer: 2580 in^2

2.
P=200cm
a=40.2cm
A=?

A=1/2P*a
1/2(200*40.2)
1/2(8040)
Answer:4020cm^2
 
TRY IT YOURSELF
 A=868
s=?
a=15.5
n=7

Answer:

s=16

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

Aim: How do we find the area of parallelograms, kites and trapezoids?
Answer:
Area of parallelograms:
    The area of a parallelogram is given by the formula which is the same as a rectangle which is b*h=a, where A is the area, B is the length of the base and h is the height of the parallelogram.

Example:


http://www.k6-geometric-shapes.com/image-files/formula-area-parallelogram.jpg


   b*h=a
                                                                                   6*5=a
                                                                                   a= 30    
http://image.tutorvista.com/content/feed/tvcs/470u93ki057suuw74ft.jpg

 Area of a Trapezoid:
                    The area of a trapezoid is given by the formula A=(b1+b2)/2*h  or A= 1/2h(b1+b2) where A is the area,b1 is the first or top base, b2 is the second or bottom base and h is the height of the trapezoid.

Example:
http://www.calculateme.com/cArea/area-of-trapezoid.gif

             A=(b1+b2)/2*h
                                                                                               A=(2.5+6)/2*5
                                                                                                A=(8.5)/2*5
                                                                                                 A=4.25*5
                                                                                                  A= 21.25
http://mste.illinois.edu/courses/mat764fa04/folders/traska/MAT%20764/Final%20Unit%20Plan/trapezoid2.jpg


Area of Kites:
        The area of a kite is given by the formula A=(d1*d2)/2, where A is the area, d1 is the first diagonal, d2 is the second diagonal.

Example:

http://www.k6-geometric-shapes.com/image-files/formula-area-rhombus.jpA
A=(d1*d2)/2                                                                 A=(8*6)/2
                                                                 A=(48)/2
                                                                 A=24
http://s2.hubimg.com/u/3564209_f260.jpg



 TRY IT YOURSELF:

What is the area of the parallelogram?


http://www.infobarrel.com/media/image/7554.jpg

What is the area of the trapezoid?







    http://www.wyzant.com/Help/Images/trap4.gif

what is the area of the kite?

http://www.bukisa.com/articles/330258_calculating-the-area-of-a-kite-shapeeasy-guide-with-diagrams



Answer:
7*9/2==> 63/2==> Answer: 31.5