When learning about area concepts, the area of regular polygons is not the place that you start. Most people are familiar with the area of rectangles, squares, and even triangles. However, the area of a regular polygon is dependent upon the knowledge of these prior concepts of area before you can fully appreciate this similar formula.
The student will investigate the concept of area.
The student will derive a formula for the area of a regular polygon.
The student will apply the area formula for a regular polygon.
Materials
Application of the formula for a Regular Polygon
When measuring the lengths of the regular hexagon, measure to the nearest whole inch. When calculating the area of the regular hexagon, round to the nearest whole inch.
Regular Polygon Check List
Use the below check list to work through the activity. Work in the order that is presented. You must complete the investigative section and determine the formular for a regular hexagon prior to applying the formula in the latter portion of the activity.
Materials
Take a sheet of paper and a compass and construct a circle as large as possible. Be sure to mark the center of the circle.
Without changing the measure on your compass, start from a point on the circle (your choosing) and continue marking additional points around the circle with the given distance on your compass. There should be a total of 6 points.
Connect the 6 points into a regular hexagon (6 sided polygon).
Connect the 6 points of the hexagon with the center of the circle. You should end up with 6 congruent circles.
Turn your focus to one of the six triangles. Find the height of the triangle measured along a perpendicular from the center vertex to the midpoint of the side opposite. What is the measure of that height.? Some textbooks will describe this height as an "apothem." An apothem of a regular polygon is a perpendicular segment from the center of the polygon's circumscribed circle to a side of the polygon.
Now that we have the measure of the height. Measure the length of the base.
Take the height of the triangle (apothem) and the length of the base of the same triangle and claculate the area of that triangle.
Are the 6 triangles congruent? If so, can you then calculate the sum total of the regular hexagon? Do we add the area of the six traingles together or can we just multiply by 6?
Let the length of each side of the polygon be called "s". Let the height of the triangle (apothem) be called "a". Let the number of sides of the polygon be called "n". So the formula for the area of a regular polygon is the area of a triangle times the number of sides of the polygon. So...what would that be? The area of one of the triangles times the number of triangles
Let's take our new formula for a regular polygon and use it. Are all baseball fields made alike? Do they all have the same outfield, foul territory, or same size dugouts. If you have been a follower of any high school baseball you know that every field is different and many are kept to any kind of standard. We are going to take a look at the pitchers mound. Gather up your measuring tape, paper, and pencil and let's go find out about your local field.
Locate what you believe to be the center of the pitching mound area. It should be a cut out area in the center of the infield. Using the measuring tape as the length of a radius of a circle, keep the tape tight and draw a circle around that center point. This is easily done with two people.
As in the beginning section of this activity, use the measure of the tape measure and mark your six points to give your self the six points of a regular hexagon. Focus on one of the triangles and measure the length of the apothem, length of a side, and then calculate to the nearest inch the area of the largest regular hexagon that will fit with in this pitching mound.
Take your data that you have gathered and share your results.
Depending on the skill level of your students, this may take more than one day. Typically, this does not take much time to grasp the concept
Cooperative learning is a good thing here. You might have the students work independently on deriving the formula. Then, you might have them work in larger groups as they go out to calculate the area of the selected places.
The formula we are exploring is A = (1/2)asn or A = (1/2)ap, where A is the area, a is the apothem, s is the length of each side, n is the number of sides of the polygon, and p is the perimeter of the polygon.
Take special note when you are calculating the area, always double check your result, even if you are using a calculator. Also, pay close attention when you are reading the measuring device.
Special Note About Area....The ACT, SAT, PASS test and other testing groups love area application problems. Area is a great life lesson, but they will have a more immediate effect on your college entrance exams.
Depending on the skill level of your child, this may take more than one day. Typically, this does not take much time to grasp the concept.
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