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Study Notes 018:
By William E. Steinman:
Geometry 11 Areas:
Let’s get simple with more math review:
A square unit is the surface area enclosed by a square whose side is one unit.
The area of a closed square figure is the number of square units contained in its surface.
The area of a rectangle is equal to the product of the length and the height of the rectangle.
A square is a special case of a rectangle wherein the length and height are equal.
Therefore, the area of a square is equal to the length squared.
We usually represent area with a capital A.
The area of a parallelogram is equal to the product of the length of a side and the height. A drawing can show this. Just cut a right triangle from one end and move it to the other end and you have a rectangle.
The area of a triangle is equal to one-half the product of the base times altitude.
This is best show with a right triangle where it can be easily seen as half of a square or rectangle.
Polygons have equal areas if they have congruent bases and altitudes.
This is also true of triangles.
A median divides a triangle into two triangles of equal areas.
A regular polygon is an equilateral and equiangular polygon.
The center of a regular polygon is the common center of its inscribed and circumscribed circles.
A radius of a regular polygon is a segment joining its center to any vertex.
A central angle of any regular polygon is the angle between two radii drawn to any two successive vertices.
An apothem is defined as the perpendicular distance from the center of a regular polygon to any of its sides. That is, it intersects the side at its center.
The perimeter of a regular polygon is the sum of the lengths of it sides or the product of the length of one side and the number of sides.
Stated another way, the perimeter of a regular polygon of sides n is equal to the length of one side times n or ns where s is the length of a side.
The area of a regular polygon equals one-half the perimeter times the length of its apothem. It is not so easy, but we can show that is true.
We can divide the polygon up into n triangles, one for each side of the polygon.
First draw the radii from the center to each of the vertices. That gives us n triangles.
Then draw the apothem from the vertex to the base of one triangle. The apothem is then the height of the triangle.
We know the area of a triangle is equal to ½ the height times the base.
So the area of a regular polygon is n(1/2rs) = 1/2nrs = ½pr where p is the perimeter, r is the apothem, s is the length of a side, and n is the number of sides.
Let me introduce the constant pi or π.
π is the ratio of the circumference of a circle to the diameter. It is a non repeating decimal number that is usually given as 3.14 or 3.1416 depending on the accuracy required.
This means the circumference of a circle can be stated as π d where d is the diameter.
We could also write 2 π r where r is the radius.
In finding the area of a circle it is useful to think of a circle as a regular polygon having an infinite number of sides. For example, we can start with a square and keep doubling the number of sides. Soon, we will not be able to discern the difference between a circle and the polygon of many sides. We reach a point where the apothem and the radius are essentially equal.
From our previous example of determining the area of a regular polygon we know that A = ½ pr
Notice that p the perimeter of a polygon is the same as the circumference of the circle which we know equals 2πr.
With simple substitution we get A = ½( 2πr )r = πr2
Therefore A = πr2.