Lesson 8-4

Areas of Regular Polygons

In this lesson you will…

● Discover the area formula for regular polygons

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PolygonPolygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up).

Regular and Irregular PolygonsIf all angles are equal and all sides are equal, then it is regular, otherwise it is irregular

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sLet’s recall some concepts

You can divide a regular polygon into congruent isosceles triangles by drawing segments from the center of the polygon to each vertex.

Center of a polygon The center of its circumscribed circle

Radii of a polygon the radius of its circumscribed circle, or the distance from the center to a vertex.

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s If you divide regular polygons into triangles. Then you will be able to write a formula for the area of any regular polygon.

To find the are of a triangle you use the following formula

𝐴=12𝑏 . h

Consider a regular pentagon with side length s, divided into congruent isosceles triangles. Each triangle has a base s and a height a.

What is the area of one isosceles triangle in terms ofa and s?

½ as

What is the area of this pentagon in terms of a and s?

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What is the area of this pentagon in terms of a and s?

5.½ as

What is the area of this hexagon in terms of a and s?

6.½ as

What is the area of this heptagon in terms of a and s?

7.½ as

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The apothem of a regular polygon is a perpendicular segment from the center of the polygon’s circumscribed circle to a side of the polygon. You may also refer to the length of the segment as the apothem.

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The apothem is the height of a triangle between the center and two consecutive vertices of the polygon.

Apothem

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The apothem of a regular polygon is a perpendicular segment from the center of the polygon’s circumscribed circle to a side of the polygon. You may also refer to the length of the segment as the apothem.

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The apothem is the height of a triangle between the center and two consecutive vertices of the polygon.

Apothem

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Hexagon ABCDEF with center G, radius GA, and apothem GH

𝑝=𝑛∗𝑠

n = number of sidess = base A

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s In a regular polygon, the length of each side is the same. If this length is (s), and there are (n) sides, then the perimeter P of the polygon in terms of n and s is:

The number of congruent triangles formed will be the same as the number of sides of the polygon.

The area of a regular polygon is given by the following formulsa

where ….A is the area, P is the perimeter, a is the apothem, s is the length of each side, and n is the number of sides.

𝐴=12ans

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𝐴=12𝑎𝑃or

Find the unknown length accurate to the nearest unit, or the unknown area accurate to the nearest square unit.

Recall that the symbol is used for measurements or calculations that are approximations.

A ?s = 24 cma 24.9 cm

A 2092 cm2

A 19,887.5 cm2

s = 107.5 cma ?

a 74 cm

P ?A = 4940.8 cm2

a = 38.6 cm

P 256 cm

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Find the approximate area of the shaded region of the regular polygon.

8.0 ft

5.5 ft

• Find the area of the entire pentagon• Find the area of unshaded triangle

• Then subtract

Another look... A = Area of 1 triangle • # of trianglesA = ( ½ • apothem • side length s) • # of sidesA = ½ • apothem • # of sides • side length sA = ½ • apothem • perimeter of a polygon

This approach can be used to find the area of any regular polygon.