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LESSON TWENTY-SEVEN:
IT’S WHAT’S ON THE INSIDE THAT COUNTS!
(PART II)
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AREA OF POLYGONS
• So far this year, we have discovered how to find the area of triangles and quadrilaterals of various size and shape.
• Today, we move on to figures with more than four sides.
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AREA OF POLYGONS
• Any figure with more than four sides can be called a polygon.
• The prefix poly- simply means “many”.• A polygon, therefore, is a figure with many
sides.
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AREA OF POLYGONS
• Like most things we discuss, it will help us greatly to first discuss the parts of a regular polygon.
• Firstly, a regular pentagon is one in which all sides and angles are equal.
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AREA OF POLYGONS
• The center of a polygon is the intersection of all the lines of symmetry of the polygon.
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AREA OF POLYGONS
• The radius of a polygon is the distance from the center to any vertex.
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AREA OF POLYGONS• The apothem of a regular polygon is the
length from the perpendicular bisector of a side to the center.
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AREA OF POLYGONS
• The central angle of a polygon is an angle whose vertex is the center and the legs extend to the polygon’s adjacent vertices.
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AREA OF POLYGONS• You may have noticed that a central angle of
that polygon makes a triangle.• Say the area of one of those triangles was 6 cm².• What would the area of the whole pentagon
be?
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AREA OF POLYGONS• It would be 30 cm² because there would be
five total triangles.• So to find the area of a regular polygon we
need to multiply ½ x base x height x number of triangles.
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AREA OF POLYGONS• Technically, the height would be the apothem
of the polygon and the base would be one side length.
• So for short, we say ½nsa• If I multiplied the n (number of sides) and s
(side length), it would be the perimeter of the polygon.
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AREA OF POLYGONS
• So, my entire equation could be expressed as A = ½ Pa.
• Using this information, you can find many things about regular polygons!
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AREA OF POLYGONS
• Given the that the figure below is a regular hexagon, find its area.
4 cm
5 cm
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AREA OF POLYGONS
• Given that the following regular pentagon has an area of 20 cm, find the length of one side.
4 cm
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AREA OF POLYGONS
• You can also find the area of a regular polygon by a method called sectioning.
• This basically calls on old knowledge of the areas of triangles, rectangles and trapezoids.
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AREA OF POLYGONS• Say we have this regular octagon for instance.• Once we have a side length and another part,
that’s all we need.
5 cm 13 cm
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AREA OF POLYGONS
5 cm 13 cm
5 cm
5 cm
5 cm
5 cm
5 cm
5 cm
5 cm
27 cm² 27 cm²
65 cm²
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AREA OF POLYGONS
• 27 + 27 + 65 = 119 cm²• So the area of the regular octagon is 119 cm²• We can do this process with other regular
polygons as well!
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AREA OF POLYGONS• Take this regular pentagon.• When we section it up, we can discover its
area as well!
17 cm
30 cm
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AREA OF POLYGONS
17 cm
30 c
m
17 cm
17 cm
17 cm
17 cm
15 cm
15 cm
8 cm120 cm² 120 cm²
8.5 cm 8.5 cm
28.8 cm
244.6 cm²
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AREA OF POLYGONS
• 120 + 120 + 244.6 = 484.6 cm²• So by sectioning, we can solve the area for
ANY regular polygon!
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AREA OF POLYGONS• Again, I will be offering you very little help on
this type of problem.• It will be your job to solve these puzzles and
find the area! • You have all the pieces you need!