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APPLICATIONS OF POLYGONS AND CIRCLES

In this unit, you will investigate some interesting applications of polygons and circles.

You will first determine the area of any given regular polygon and explore what happens

to the shape of a polygon as the number of sides increase. You will also examine the area

formula and how it is derived from an infinite number of sides in a regular polygon. You

will then look at geometric probability based on length and area. The final part of this

unit is investigating graph theory which is a study of networks that connect nodes with

straight or curved paths.

Area of Polygons

Area of a Circle (Derivation)

Geometric Probability

Graph Theory

Area of Polygons and Circles

apothem - An apothem is a perpendicular line segment that is drawn from the

center of a regular polygon to its side.

If the length of the apothem and perimeter of a regular polygon is known, then

the area can be calculated.

Area of a Regular Polygon

For a regular polygon with an area of A square units, a perimeter of P units,

and an apothem of a units, the area equals 1/2 the perimeter times the

apothem.

A = 1 Pa

2

Let's examine how this formula is derived.

S

R T

Regular Hexagon QRSTUV is inscribed in Circle W.

Segments RW, SW, TW, UW, VW, and QW are radii. W

Segment WX is an apothem. X

Q U

V

TWU is isosceles. WT and WU are congruent radii.

TWU , UWV , VWQ, SSS (Sides are congruent radii and bases are

QWR, RWS, and SWT congruent segments of the regular hexagon.)

are congruent

The area of the hexagon The area of a polygon equals the sum of the areas

equals the sum of the area of its non-overlapping regions.

of the six triangles above.

Area of TWU = 1 (TU)(WX) Formula for the area of a triangle (A = 1 bh).

22

Let s represent TU , the base of triangle TWU (a side of polygon QRSTUV) and a

represent WX , the apothem.

How do you calculate the area of shaded regions? How to Find the Area of a Shaded Region? Firstly, find out the area of the large geometrical shape or outer region. Then, find the area of the small geometrical shape or inner region of the image. Finally, subtract an area of the small geometrical shape (entire region) from the large area of the small geometrical shape (shaded region). More items...

Title: Graph Theory

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