Objectives:
Students will be able to
1) define polygonal region
2) find the perimeter of a polygon
3) determine the area of parallelograms (including the rectangle, rhombus, and square), triangles, trapezoids, and regular polygons.
A polygon is a plane figure whose sides are three or more coplanar segments that intersect only at their endpoints. Consecutive sides cannot be collinear and no more than two sides can meet at any one vertex.
A polygonal region is defined as a polygon and its interior.
The perimeter of a polygon is the distance around the figure. The area of a polygonal region is measured in square units. For every polygonal region, there is a unique positive number called the area of the region.
The area of a polygonal region is the sum of the areas of all its non overlapping parts. The area of a rectangle can be found by counting the number of square units inside the region. In the rectangle below, there are 18 square units of area within the figure.
Another way to find to find the area of a rectangle is to multiply the number of units along the base times the number of units of height, or simply Area = bh. In the above rectangle, the area would be calculated by multiplying 6 X 3 to get 18 square units.
Other useful area formulas that are demonstrated in geometry textbooks are shown below. In each case, there must be an appropriate height that is perpendicular to the base.
Studying the area formula of a rectangle, square, parallelogram (rhombus), triangle, and trapezoid will enable students to calculate many areas.
AREA PRACTICE
Use the appropriate base and height in the figure to calculate area:
Click here to check your work.
Finding perimeter is simply summing the lengths of the sides.
Find the perimeter of the following figures.
Click here to check your perimeters.
Some figures might be shaped in a combination of figures explored thus far. Since the polygonal region is the sum of non-overlapping regions, several formulas for area might be used to get a total area.
Area Practice for Various Polygonal Regions
Find the area of the following figures:
Click here to check your answers.
Another type of polygon and its region that can be explored is the regular polygon. A regular polygon is both equilateral and equiangular. The center of a regular polygon is the point of intersection of the perpendicular bisectors of the sides. The segment from a vertex to the center is the radius. The perpendicular segment from the center to a side is an apothem. The picture below illustrates these definitions.
The area of a regular polygon is one-half the product of its perimeter and its apothem, or A=(ap)/2. The picture below illustrates finding the area to an equilateral triangle:
Students should be able to find the area of various regular polygons if they have enough information about the figure, such as angles measures, length of the center, the radius, or the length of a side.
PRACTICE ON AREA OF REGULAR POLYGONS
From the given information, find the area of the regular polygons below.
This concludes lesson # 1 on area. You should now be able to find the area of various types of geometric figures.
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