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Decomposing Polygons to find Area and more ...



Decomposing (or decomposition) is the process of dividing a geometric figures into smaller (non-overlapping) recognizable shapes. The general purpose of decomposing is to have shapes within the given figure that can be more easily manipulated for computations.

We have already used this concept in situations such has finding the area of challenging figures, and discovering the sum of interior angles in a quadrilateral.
Note: the "refresher" examples below reflect our previous work with decomposing.

Refresher: Decompose this trapezoid
and find its area.
ptrap
Solution: The figure has been decomposed into two triangles and a rectangle. The area of the rectangle is 14 x 9 = 126 sq. units.
The area of each right triangle is ½(5)(9) = 22.5 square units.
Total area is 126 + 22.5 + 22.5 = 171 sq. units
(The formula for the area of a trapezoid was not needed.)

Refresher: Find the sum of the measures of the interior angles of a quadrilateral by decomposing the figure.
pquad
Solution: A diagonal was drawn on the quadrilateral dividing it into two triangles. Since the sum of the measures of the angles of a triangle is 180º, the sum of measures of the angles of a quadrilateral must be 360º (twice 180º), since it is made up of 2 triangles.

We will continue to use decomposition (and other) strategies when we work with polygons with more than four sides. Consider these examples:
The decompositions used in these examples may not be the only possible patterns.

ex1 Find the area of this regular hexagon which has been decomposed. Round answer to nearest tenth of a sq. unit.


Solution:
A regular hexagon has 6 sides of equal length.
The hexagon has been divided into 4 small right triangles and one rectangle.
• The area of the 4 right triangles is ½(5.196)(3)=7.794 sq. units for each triangle

• The area of the rectangle is 2(5.196)(6)=62.352 sq. units.

Total area = 4(7.794) + 62.352 = 93.528 = 93.5 sq.units

hexarea
hexareanote

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ex2 Find the sum of the measures of the interior angles of a pentagon.

Solution:
The same strategy that was used with the quadrilateral in the refresher above, will be used here.
A pentagon has 5 sides. Draw as many diagonals as possible
from the same vertex, to decompose the pentagon into three triangles.
A pentagon is now composed of three triangles, each having the sum of their angles = 180º.
The sum of the interior angles of a pentagon will be 3 x 180 = 540º.

ppent

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ex3 Find the area of this regular octagon which has been decomposed.


Solution:
The regular octagon (with all 8 sides of = length) is divided into 5 rectangles and 4 triangles.
• Each of the small Δs has an area of ½(4.4)(4.4) =
9.68 sq. units each.
• The square in the center is 6.2 x 6.2 with an area of 38.44 sq. units.
• The 4 remaining rectangles are 4.4
x 6.2 with an area of 27.28 sq. units each.
The total area is
4(9.68) + 38.44 + 4(27.28) = 186.28 square units.



9ctagonarea
Lengths have been given to nearest tenth.

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ex4 Find the number of degrees in each interior angle of a regular hexagon.


Solution: In Example 2, we saw how to decompose the pentagon into triangles by drawing diagonals from one vertex. Using this same strategy in a 6-sided hexagon, produces 4 triangles. 4 x 180 = 720º.
The sum of the measures of the angles in a hexagon is 720º.
Because this is a "regular" hexagon, we know that all of its angles are of the same measure.
If we divide 720 by 6, we find that
each interior angle has a measure of 120º.

hexregular

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ex5 Find the area of this irregular pentagon which has been decomposed.


Solution: This irregular pentagon has been decomposed into a square and a triangle. While not ALL of its sides are the same length, the hash marks tell us that 3 of them are the same length.
• The area of the square is 15
x 15 = 225 sq. units.

• The are of the triangle is ½(6)(15) = 45 sq. units.

The total area = 225 + 45 = 270 sq. units.

pentirr



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