LESSON TWENTY-SEVEN:

IT’S WHAT’S ON THE INSIDE THAT COUNTS!

(PART II)

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.

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.

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.

AREA OF POLYGONS

• The center of a polygon is the intersection of all the lines of symmetry of the polygon.

AREA OF POLYGONS

• The radius of a polygon is the distance from the center to any vertex.

AREA OF POLYGONS• The apothem of a regular polygon is the

length from the perpendicular bisector of a side to the center.

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.

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?

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.

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.

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!

AREA OF POLYGONS

• Given the that the figure below is a regular hexagon, find its area.

4 cm

5 cm

AREA OF POLYGONS

• Given that the following regular pentagon has an area of 20 cm, find the length of one side.

4 cm

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.

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

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

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!

AREA OF POLYGONS• Take this regular pentagon.• When we section it up, we can discover its

area as well!

17 cm

30 cm

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

AREA OF POLYGONS

• 120 + 120 + 244.6 = 484.6 cm²• So by sectioning, we can solve the area for

ANY regular polygon!

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!