Area and Perimeter (Pikachu)

pikachu (1)Examples come from RegentsPrep.org

Practice Problems:

Perimeter and Circumference

Area of Polygons and Circles

Area on a Coordinate Grid

 

 

Perimeter is the word used to describe the distance around the outside of a figure.

To find the perimeter,
add together the lengths of all of the sides of the figure.

 

Triangle 3 sides
Quadrilateral 4 sides
Pentagon 5 sides
Hexagon 6 sides
Heptagon or Septagon 7 sides
Octagon 8 sides
Nonagon 9 sides
Decagon 10 sides
Dodecagon 12 sides

Refresh your polygon memories:

When working with perimeter, references may be made to the names of polygons.  Listed at the left are some of the more common polygons whose names you should know.

Remember that “regular polygons” are polygons whose sides are all the same length and whose angles are all the same size.  Not all polygons are “regular”.

Circumference is the word used to describe the distance around the outside of a circle.

Like 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.

 

Example 1:

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.


= (3.141592654)(0.8) = 2.513274123 miles (one trip)
2(2.513274123) = 5.026548246 = 5 miles
This problem is shown being done with full calculator (TI-83+/84+) entries.
Work should be held in the calculator until the final rounding of the answer occurs.

Example 2:

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

Remember to label areas with “square” units.

 Area (triangle)

 Area (equilateral triangle)

or

 Area (rectangle)

or

 Area (rectangle) = (length)•(width) 

 

 Area (square)

 Area (parallelogram

 Area (trapezoid

d1=diagonal 1
d2= diagonal 2

 Area (rhombus)

or

 

 Area (circle)

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


Semi-circle

(half of circle = half of area)


Quarter-Circle
(1/4 of circle = 1/4 of area)


Any Sector
(fractional part of the area)
(optional topic)


where n is the number of degrees in the central angle of the sector.


where CS is the arc length of the sector.

                      

This area formula is an optional topic.  Perimeter of regular polygons, however, is a required topic.

Area (regular polygon)

 

Regular polygons have all sides of equal length.

a = apothem
p = perimeter

Area formulas can be found at “Reference Table for Areas“.

Let’s pick up some hints for those more challenging problems involving area …

1.

Find the area.

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

2.

Find the area of
parallelogram ABCD

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.

3.

Find the 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.

4.

Find the area of the circle.  Round answer to nearest tenth.

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.

5.

Find the area of this trapezoid.

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.

 

 

 

6.

Find the area of the rectangle.

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

 

Finding the area of polygons drawn on a coordinate axis is an easy process.   There are two situations to be considered when examining these polygons:

Sides are parallel to the axes.

If the figure is drawn such that its sides (or needed segments) are drawn ON the grids of your graph paper, you can COUNT the lengths and use your area formulas.

Sides are NOT parallel to the axes.

If the figure is drawn such that its sides (or needed segments) are NOT drawn ON the grids of your graph paper, you will need to draw a “BOX” around the figure to determine its area.

Sides are parallel to the axes:

 

COUNT
to find the needed lengths.

In this example, the base of the triangle lies onthe 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.

The answer is 9 square units.

Sides are NOT parallel to the axes.

 


** Find the area of the “box” by counting.** Represent the triangle you wish to find by x.

“Box” Method
to find area.
In this example, the sides of the triangle doNOT 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 the area of each of the right triangles by
counting and using the formula for the area of a triangle.
       
The answer is 12 square units.

 

 


Dealing with odd shaped pieces.

Further subdivide …
There are times when the parts of the “box” will not form nice right triangles in each of the corners.  Notice the upper left hand corner of the “box” in this example.  It was necessary to further subdivide that upper left section into one square and two right triangles, so that the lengths could be counted easily..

Remember to keep your work simple by forming shapes that are easy to count.

Follow the same procedure that was developed in the previous problem.


The answer is 33 square units.

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