math report final

AREAS OF GEOMETRIC

FIGURES

POLYGON SONGRefresh your polygon memories:

REFERENCE TABLE OF GEOMETRIC FIGURES

REGULAR POLYGONS

I. Exercises on Polygons:

Be careful of problems that give "extra" information.  In this problem, the 24 is NOT needed to compute the area.

When working with parallelogram problems, be sure that the height you are using is in fact perpendicular (makes a right angle) to the base (side) you are using.  In this problem, 8 is the base and 9 is the height.  The side of 10 is not used in this area.

It may be necessary, when working with an obtuse triangle, to look outside the triangle to find the height.  Notice how the height is drawn to an extension of the base of the triangle.

When working with circles, be sure that you are using the radius.  In this diagram, 10 is the diameter.  The radius is half of the diameter.

When working with a trapezoid, the height may be measured anywhere between the two bases.  Also, beware of "extra" information.  The 35 and 28 are not needed to compute this area.

Some problems may require that you find an additional piece of information BEFORE finding the area.  This problem expects you to use the Pythagorean Theorem to find the base of the rectangle BEFORE finding the area

CIRCUMFERENCE OF CIRCLESLike perimeter, the circumference is the distance round the outside of the figure.  Unlike perimeter, in a circle there are no straight segments to measure, so a special formula is needed.

Use when you know the radius.

Use when you know the diameter.

Ed and Carol are jogging around a circular track in the park.  The diameter of the track is 0.8 miles.  Find, to the nearest mile, the number of miles they jogged if they made two complete trips around the track.

Example 1:

 (3.14)(0.8)5.026548246 = 5 miles

For an art project at school, you need a piece of string long enough to wrap around the outer edge of this starfish.  What is the shortest possible length for the string?

Perimeter = 2 + 1.5 + 1 + 2 + 1.5 + 2 + 2 + 3 + 2.5 + 2 = 19.5 inches

Sectors and Segments of Circles

Area (circle)

Example#2

    Area  of sectors of circle          (Sectors are similar to "pizza pie slices" of a circle.)

Notice that when finding the area of a sector, you are actually finding a fractional part of the area of the entire circle.  The fraction is determined by the ratio of the central angle of the sector to the "entire central angle" of 360 degrees, or by the ratio of the arc length to the entire circumference.  The second formula can be algebraically reduced, but it is easier to remember that you are dealing with fractional parts.

EXAMPLE#1Find the area of a sector with a central angle of 60 degrees and a radius of 10.  Express answer to the nearest tenth. Solution:

EXAMPLE#2Find the area of a sector with an arc length of 40 cm and a radius of 12 cm. Solution:

Area of polygons on a coordinate axis

Sides are parallel to the axes: COUNT 

to find the needed lengths.In this example, the base of the triangle lies on the grid of the graph paper, and the altitude also lies on the grid of the graph paper.To COUNT:  stand at A and take one step to the right to the next grid line.  Continue stepping and counting until you reach C.From counting, we know the base is 6 and the altitude is 3.The area of a triangle formula:

You could also find the length from A to C by subtracting the x-coordinates of the two points.    4-(-2) = 6For the altitude, you need to determine that the base of the altitude is (2,1).  Then subtract the y-coordinates of the two points.   4 - 1 = 3.

Sides are NOT parallel to the axes.

"Box" Method to find area.In this example, the sides of the triangle do NOT lie on the grid of the graph paper.  You should:1.  Draw the smallest "box"

possible which will enclose the polygon (in this case a triangle).  Be sure the "box" follows the grids of the graph paper.2.  Number each of the parts of the box with a Roman numeral (ignore the coordinate axes when numbering). 3. "The whole is equal to the sum of its parts."  The area of each of the parts added together equals the area of the box.

** Find the area of the "box" by counting.

** Represent the triangle you wish to find by x.

** Find the area of each of the right triangles by counting and using the formula for the area of a triangle. 

It's easy, but it's tiring!

END OF MY REPORT

Reported by: Ronald A. Sato

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