Chapters 12 & 13: Prisms and Pyramids vs. Cylinders and Cones

Prisms: SA= 2B + Ph
              V= Bh
Pyramids: SA= B + 1/2Pl or SA= B + LA     (LA= 1/2Pl)
                  V= 1/3Bh or V= Bh/3
Cylinders: SA= 2(pi x r squared) + (2 x pi x r)h
                  V= pi x r squared x h
Cones: SA= pi x r squared + LA     (LA= pi x r x l)
              V= 1/3 x pi x r squared x h or V= pi x r squared x h/3

[SA= surface area, B= area of the base, P= perimeter of the base, h= height, V= volume, l= slant length, LA= lateral area, r= radius (the x is a multiplication sign, not a variable)]

The formulas for these 3-D shapes can be compared and contrasted by looking at many different qualities. First of all, each volume formula involves having to find the area of the area of the base of the object. Both cylinders and cones implement the use of pi because their bases are circles; however, any shape that does not contain a circle will not need to use pi in its formula. To find the volume of cones and pyramids, one must multiply by one-third (or divide by three) to find the value because each shape is a third of either the prism or cylinder. Also, all of the formulas use height, or slant height for cones and pyramids. Finally, the formulas are different because each has to conform to the different angles and properties of the shapes; for instance, since the bases of cones and cylinders are circles, one's volume and surface area measurements will not be exact. This is because pi is infinite and does not terminate. This, however, does not apply to prisms or pyramids because they are not involved with circles. These are only some similarities and differences between these formulas and their shapes.

Anne Isabella Byron-Milbanke

This great woman of mathematics had a discouraging life. She married Lord Byron on January 2nd, 1815. This man would eventually become an untreatable affliction in this woman's life. He also treated her with little to no respect. Even though he affectionately nicknamed her "The Princess of Parallelograms", he had multiple affairs and led a morally fractured life-style. She, however, led a very productive life; science, math, religion, and philosophy all had a significant influence on her. She was a brilliant woman plagued with the constant attention of this fearful man.
In March, 1816 the couple separated after Lord Byron began exhibiting odd and sometimes insane actions. They had one child named Ada who was also raised in the strict, morally correct up-bringing that Annabella was. She also was brilliantly gifted and showed much potential.
Lady Byron, "The Princess of Parallelograms", died from breast cancer on May 16th, 1860 the day before her 68th birthday. The woman who contributed so much to society was as much in need of society as society was in need of her.

Chapter 11: Area of Parallelograms

After today's lesson, I found out that the formulas to calculate the area of a square/rectangle is the same as any parallelogram. This is because all parallelograms are just rectangles with triangles placed on the ends. The formula can either be written A=length x width or A=base x height.
Tennessee is a good example on how to find the area of a parallelogram. Since it is roughly a parallelogram, we can apply this formula to it. Tennessee's dimensions are as follows: length=440 miles and width=120 miles approximately. The area of Tennessee can, therefore, be calculated by multiplying 440 by 120.
A=440 x 120 ---> A=52,800 miles squared.
This gives us a fairly rough estimate of the area of Tennessee. The actual area is 42,143 miles squared. The difference is about 10,000 miles which is pretty large; however, it is an educated guess and a reasonable way to find the answer.

Chapter 10: Circles

I have learned much from our study of circles in Chpter 10. I have learned that a tangent is a line that intercepts a circle at one point, that inscribed angles are half of their intercepted arc and that central angles are equal to their corresponding arc measurements. Many jobs can apply geometry into their everyday working routine. For example, statisticians must know what percentages and angle measures a pie chart must equal for it to equal 360 degrees and for it to equal 100%. I am still wondering how it is possible for some vertical angles to not equal each other when intercepted in a circle. On many problems, I have noticed that opposite arcs are not equal and I'm wondering how this is happening and how this can be explained. Help!