In this lesson youll learn the basics and concepts of Area of Polygons and Circles, an important part of the geometric applications. First the related and earlier learnt basics will be reviewed, which will follow on the important relationship of parts of the polygons/circle and angular measurements. and that help to apply them to solve real world application problems. The presentation covering such content will be done by the instructor in own handwriting, using video and with the help of several examples and solution. This will help you understand important trigonometric relationships to solve home work problems and also use them for real life applications.
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Area is a measure of the amount of space contained inside a closed figure i. e. polygon and circle. The area of a regular polygon may involve relatively simple figures such as an equilateral triangle, square or complex figures combining different shapes. To find the area of say an equilateral triangle, you can use the formula:1/2
* (bh), where b is the base and h, altitude or height of the triangle. Similarly area of
a square with side 's', shall be s^{2} sq units.
Notice that in cases where the number of sides of the regular polygon is 3 or 4, it may be easy to calculate the area. Similarly area of a regular hexagon also is simple to calculate, as it can be broken into 6 equilateral triangles. Then sum up for all the equilateral triangles. Now look into a case of a regular polygon with n number of sides. You may recall the required trigonometric relationships, while working out the formula for determining the area:
In the figure on right, look at the part of the regular polygon with side s and n number of sides. Point
O is the center of the polygon and r, the distance from the center to a vertex i.e. radius of the polygon.
With this information, formula for the area can be derived:
POQ
has a measure of (360/n),
and MOQ
has a measure of (180/n).
POQ
is a central triangle, and there shall be n such triangles in this polygon.
Since PQ is side with measure s,
MQ = s/2. (basic right triangle trig)
Now that,
h = r cos(180/
n) and b = r sin(180/
n)
and MQ = s/2
Thus area of
POQ = 1/2 (b* h)
= s/2 * h = sin(180/
n) * cos(180/
n) r^{2}
This is the area of POQ.
Notice that the polygon has n number of such triangles, area of the polygon shall be:
The area of a regular polygon of n sides and radius r = n * sin(180/
n) * cos(180/
n) r^{2
}
Common Formulas for Area
Wikipedia
Shape
Formula
Variables
Square
s is the length of one side of the square.
Rectangle
l and w are the length and width of the rectangle.
Rhombus
a and b are the lengths of the diagonals of the rhombus.
Parallelogram
b is the length of the base and
h is the perpendicular height.
Equilateral triangle
s is the length of one side of the triangle.
Triangle
a, b and c are the length of each side, and s is half the perimeter.
Triangle
a and b are any two sides,
and is the angle between them.
Triangle
b and h are the base and altitude (measured perpendicular to the base), respectively.
Trapezoid
a and b are the parallel sides and h the distance (height) between the parallel sides.
Regular hexagon
s is the length of one side of the hexagon.
Regular Polygon
a is the apothem, or the radius of an inscribed circle in the polygon, and
p is the perimeter of the polygon.
Circle
r is the radius and
d the diameter.
Circular sector
r and are the radius and angle (in radians), respectively.
Ellipse
a and b are the semimajor and semiminor axes, respectively.
Further on, youll explore more details about the area of polygons and circles looking at some of the illustrative examples to have better understanding of the concepts and relationships:
Polygon Regular A polygon with all sides and interior angles the same. Regular polygons are always convex i.e. all interior angles less than 180°, and all vertices 'point outwards' .
Properties Summary:
Sum of Interior Angles [180 (n2)], where n is the number of sides
For a regular polygon, all the interior angles have the same values. E.g. the interior angles of a hexagon always add up to 720°, so in a regular hexagon, each one is one sixth of that, or 120°. It can be thus said: each interior angle of a regular polygon is given by: [180(n2)/n], where n is the number of sides.
The apothem of a polygon is a line from the center to the midpoint of a side. This is also the, radius of the incircle.
The radius of a regular polygon is a line from the center to any vertex. It is also the radius of the circumcircle of the polygon.
Quadrilateral:
It is a polygon with four 'sides' or edges and four vertices or corners.
Quadrilateral types: Parallelogram opposite sides parallel. Rectangle opposite sides equal, all angles 90°. Square all sides equal, all angles 90°. Trapezoid two sides parallel. Rhombus opposite sides parallel and equal.
Parallelogram
It is a quadrilateral with two pairs of parallel sides.
Properties and important points to remember
Opposite sides of a parallelogram are equal in length.
Opposite angles of a parallelogram are equal in measure.
Opposite sides of a parallelogram can never intersect.
The diagonals of a parallelogram bisect each other.
Consecutive angles are supplementary, add to 180°.
Area (A), of a parallelogram is A = bh, where is the base and is height of parallelogram.
The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
A parallelogram is a quadrilateral with opposite sides parallel and congruent.
Notice that:
A rectangle is a parallelogram, but with each angle as 90°.
A rhombus is a parallelogram, but with all sides equal in length.
A square is a parallelogram but with all sides equal in length, and each angle at 90°
Example 1: Given let ABCDEF be a regular hexagon, as shown in the figure below (NTS). What is the ratio of the area of the triangle
ACE to that of the hexagon ABCDEF?
Step 1: Let the side of a regular hexagon be 'a'
Step 2: The area of hexagon ABCDEF = ((33)/2)a^{2}
Notice and consider the rhombus COED,
Step 3: Area of Rhombus
COED = (1/2) times product of the length of its diagonal = (1/2) * CE * OD = (1/2) * CE * a .................(1)
Also, Area of Rhombus COED = Area of COD + Area of DOE
= (3/4)a^{2} + (3/4)a^{2}
= (3/2)a^{2}.................(2)
Step 4: Comparing (1) and (2), gives, (1/2) * CE * a = (3/2)a^{2} i.e. CE = (3)a
Step 5: Area of ACE = (3/4)CE^{2} = (3/4)((3)a)^{2} = ((33)/4)a^{2}
The ratio of the area of the triangle ACE to that of the hexagon ABCDEF is
=((33)/4)a^{2} / ((33)/2)a^{2} = 1/2, as the final answer.
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