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

Spheres

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

S.A.= 4pi r squared

Calculate the surface area:

SA= 4 pi r squared

SA= 4pi(18) squared

SA= 1,296pi squared

V= 4 pi r cubed/3

Calculate the volume:

V= 4 pi r cubed/3

V= 4 pi (8.7) cubed/3

V= 2,634.012pi/3

V= 878.004pi cm cubed

1st sphere

2nd sphere

How do we find the volume of pyramids and cones?

How do we calculate the volume of a cone?


V= Bh/3 
B= Area of Base = pi r squared 

Calculate the volume of the cone:
V= Bh/3                       B= pi 3 squared = 9pi
V= 9pi(11)/3
V= 99pi/3
V= 33pi cm cubed

     

How do we calculate the volume of a pyramid?
V= Bh/3

Calculate the volume of the pyramid:
V= Bh/3                                  B= (4)(3) = 12
V= 12(4)/3
V= 48/3
V= 16 cm cubed



1st cone
2nd cone
1st pyramid
2nd pyramid

How do we find the volume of prisms and cylinders?

Rectangular Solid

                          

                                                 

                                                 V= Bh

 B= Area of Base

 V= 18 x 4
 V= 72 cm cubed 




Find the volume:


V= 20 x 3
V= 60 cm cubed 





B= (b1 + b2/ 2)h         
B= (12 + 6/ 2)4
B= 36 sq cm

V= Bh

V= 36 x 10
V= 360 cm cubed 



1st rectangular prism
2nd rectangular prism
trapezoidal prism

How do we do compositions of transformations?

Composition of Transformations

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

                     Do second → rx-axis º T(3,4) ← Do first

Example 1: Find the coordinates of the image of A(2,4) under the transformation ry-axis º T(3,-5).

Answer: A(2,4)→A'(5,-1)→A''(-5,-1)

Example 2: Find the image of point A(3,-2) under the composition of translation T(2,1) º T(-6,-4).

Answer: A(3,-2)→A'(-3,-6)→A''(-1,-5)

How do we define circles?

What is a circle?

A circle is the set of all points in a plane at a given distance from a given point. The given distance is a radius and the given point is a center.

Chord

A line segment that connects two points on a circle

Diameter

A line segment that goes through the center of a circle with endpoints on the circle. It is also the longest chord in the circle.

Tangent

A line segment that touches exactly one point  on a circle. Two circles can be a tangent at the same line and point.

 

Arc

A portion of the circumference of a circle.

http://www.math-worksheets.info/2010/04/circle-continued/

http://paulgoddard.co.uk/tuition/resources/mathematics/geometry/circle_facts.html

http://ef004.k12.sd.us/ch10notes.htm  

https://sites.google.com/site/geometry4sage20112012/home/resources/star-lessons/circles-ii http://jwilson.coe.uga.edu/emt668/EMAT6680.2003.fall/Nichols/6690/Webpage/Day%202.htm 

How do we calculate distances?

Distance formula:

Ex:

Calculate the length of a segment whose endpoints are at (-1,5) and (7,3).

X_2= 2, X_{1= -1,} Y_{2= 3,} Y_1= 7

d= √(2--1)² + (3-7)²
d= √(9 + 16)
d= √25
d= 5

http://booknerd3.deviantart.com/art/Distance-Formula-208719112

How do we use a polygon's exterior angles?

Exterior Angle Sum Conjecture

For any polygon, the sum of the measure of a set of exterior angles is 360°.

Regular - Equilateral, Equiangular

Ex 1: 

What is the measure of an exterior angle of an equiangular pentagon? 

Solution:

Pentagon= 5 sides 

Sum of exterior angles= 360°

measure of each exterior angle= ?

So, 360 ÷ 5(# of sides)= 72°

Ex 2: 

How many sides does a regular polygon have if each exterior angle measures 24°?

Solution:

# of sides= ?

Sum of exterior angles= 360°

measure of each exterior angle= 24°

So, 360 ÷ 24 = 15, 15 sides 

http://www.geom.uiuc.edu/~dwiggins/conj09.html