Geometry - Mrs. Saatkamp: April 2009

Wednesday, April 29, 2009

Class of Tueday, 28 of April

Today we started by learning how to solve for the area of regular polygons.

First, we solved for the area on the pentagon, on the sheet of paper that she gave us. Then, we solved for the area of the octagon, when she told us that the raduis of the circumscribed circle was 10. To solve:

y= i/2

a= apothem

x= side/2

(circumscribed circle means a cirle on the outside of the octagon that touches all the vertices of the octagon)

That means that the radius is one of the sides of a triangle made from the diagonals.

First, you have to calculate the sum of interior angles to find one interior angle, in order to find angle y:

si= 180(8-2) = 1080

i= 1080/8 = 135

y= 135/2 = 67.5

Now, we calculate the apothem:

sin 67.5= a/10

a= 9.238

Then we calculate x:

cos 67.5= x/10

x= 3.826 (remember always to use at least 3 decimal places)

Then calculate the base (side of the octagon):

b= 2x= 7.654

Then calculate the perimeter:

p= 8b = 61.229

then calculate the area:

A= (pXa)/2

A= (61.229X9.238)

A= 282.8u²

So here are the Formulas for the Area of Regular Polygon:

p= perimeter
a= apothem

(Area of Circles):

Monday, April 27, 2009

27/04/2009 - 9B Class

First we started by reviewing the equations to find the area of Polygons:

Area of the parallelogram:

Area of the triangle:

Area of the Rhombus:

Area of the trapezoid:

After that we corrected the homework that was on page 539 # 11-26

Subsequent to correcting the homework Mrs. Saatkamp gave us a little piece of paper that had three pictures of polygons, one hexagon (6 sides), one pentagon(5 sides), and one octagon(8 sides).
So we chose the hexagon to start with. We drew only the diagonals that pass through the center (3). So 6 equilateral triangles were formed and we chose one to work with. Than the height of this triangle we started to call apothem.

APOTHEM - If we can divide into N numbers of triangles, apothem will be the height of one of those.

Then we called the base of the triangle base, which is not only the base of teh triangle, but also a side of the polygon.

So we calculated the area of the triangle and multiplied by six (because a hexagon has six triangles inside of it) and what we got was:

In which (p) means perimeter and (a) apothem...

So this will be always the equation to find the area of any kind of regular polygons!!!

So remember always to use SOH CAH TOA to find the measure of the heights and bases, and try to practice it without a calculator, by using the SOH CAH TOA table, because it will be required in most of our quizzes and tests.

9"A"- 27/04/09

Today in class we started of by correcting the h.w which was pg. 539 # 11-26. We asked to solve some on the bord so I'll give an example of 2 of them.

19)

A= 95, find the value of x.

The formula for the area of a trapezoid is

so we substitute and get

then multiply 95 x 2 and the answer is divided by 2 and we get 10.

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21) A rhombus has an angle measure of 120, and its longer diagonal has a lenght of 10 inches. Find the AREA of the rhombus.

After drawing all the given information we can apply SOH CAH TOA to find the area of 1 triangle. Tan 60º= 5/x

after we simplify we get

after this we just have to substitute in the formula for area of a triangle and times the answer by 4 to get the area of the rhombus.

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After finishing with the h.w we continued to AREAS OF REGULAR POLYGONS. Mrs. Saatkamp gave us a print out with 3 figures: 1 hexagon, 1 pentagon and 1 octagon. We had to draw the diagonals in order to split the shape in triangles. With this we were going to come up a formula that works to find the are of all regular polygons.

Keep in mind-

The base of 1 triangle is 1 side of the polygon.

The hight of a triangle is a apothem.

An apothem is 1/3 of the hight of the polygon.

Hexagon: All 5 triangles are equilateral.

To find the area of the polygon without drawing it and the triangles inside it we need to get one of the triangles and apply the formula for area of a triangle BUT replace hight by apothem and base by side. To finish the formula should be multiply by the total number of sides in the polygon.

The angle of 1 of the triagles is = interior angle/2

Formulas:

-Area of a regular polygon

p= perimeter

a= apothem p= perimeter

-Area of a circle = r^2π

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We have no h.w for tomorrow yey!!

see you in class :]

Malena

Sunday, April 26, 2009

Class of April 24th

So...
Friday’s class started with Ms. Saatkamp warning us that in this last quarter there will be some quizzes WITHOUT CALCULATORS, so we better start practicing how to work with square roots and sines, cosines and tangents.
By the way, don’t forget the SOH CAH TOA Table!

After that we went on to the new subject: AREAS of TRIANGLES, RHOMBUS and TRAPEZOIDS.
TRIANGLE ~ To find out the formula for the area of a triangle, you first must remember that a triangle is half of a rectangle; therefore, the area of a triangle will be half of the area of a triangle.

RHOMBUS ~ To find the formula for the area of a rhombus, you have to keep in mind that a rhombus is also like two triangles; therefore, it is by adding both triangles areas that we will come up with the rhombus area.

Purple triangle & Black triangle

In order to find the area of each triangle,

you must use the following formulas:

TRAPEZOID ~ To find the area of a trapezoid you must also remember that it is like finding the area of two triangles.

The formulas are the following:

The homework for Monday is: Page 539, # 11 - 26 all!

Don't forget that also on Monday there is going to be a RECUPERATION TEST on CHAPTER 9!

Thursday, April 23, 2009

Class Today

Today we began class by correcting the homework p.519 # 22-49. The problems below are problems that she explained on the board:

38) The number of sides of a regular polygon is given. Find the measures of an interior angle and an exterior angle for each polygon. Round to the nearest hundredth.

n= number of sides

e = (exterior angle)

i = (interior angle)

n=3m

i = 180 - e = 180 - 120 m = 60 m

45) The measure of an interior angle of a regular polygon is given. Find the number of sides.

i =

(because this is the formula to find the number of sides, you know that the s given in the problem actually represents n, or the number of sides that the polygon has. You just have to do the distributive property)

48) If the exterior angle of a regular polygon measures 36, find the sum of the measures of the interior angles.

e=36

Si= 180 (n - 2)

36 (exterior angle) =

n = 10 ( now substitude n with ten in the formula: Si = 180 ( n - 2) and you get the interior angle measurment.)

The Lesson today: Areas and Perimeters of: Squares, Rectangles, and Parrallelograms.

Squares:

P= 4l units

Rectangles:

P = 2l + 2w units

A = lw

Parralelograms:

P = 2b + 2m

A = bh

The red triangle that is part of the parralelogram is congruent to the green triangle outside of the polygon. Without the red triangle and with the green one, the area is still the same, but now can be measured as a rectangle's area, therefore it is the height of the polygon times its base.

The homework she assigned for tomorrow is: p. 532 # 10 - 24 all

*Jenni

Today's Class

Today we started the class correcting the homework from page 519 # 22-49 all. After that we started the content of areas and perimeters some figures. The figures that we saw were the square, the rectangle and the parallelogram.