11.2 Areas of Regular Polygons - Montgomery …...10.3 Areas of Regular Polygons Geometry Mr. Peebles Spring 2013 Bell Ringer • Find the area of an equilateral triangle with 6 inch

10.3 Areas of Regular Polygons

Geometry

Mr. Peebles

Spring 2013

Bell Ringer

• Find the area of an equilateral triangle with 6 inch sides. PLEASE LEAVE YOUR ANSWER IN RADICAL FORM. A = ¼ s2 3 Area of an equilateral Triangle

Bell Ringer

• Find the area of an equilateral triangle with 6 inch sides. PLEASE LEAVE YOUR ANSWER IN RADICAL FORM. A = ¼ s2 3

A = ¼ 62 3

A = ¼ • 36 3

A = • 9 3

A = 9 3

Area of an equilateral Triangle

Substitute values.

Simplify.

Multiply ¼ times 36.

Simplify.

Daily Learning Target (DLT) Tuesday January 15, 2013 • “I can remember, apply, and understand

to find the perimeter and area of a regular polygon.”

Assignment: 10-3

Pages 548-551 (4-9, 11, 24)

4. 2144.475 cm2 9. 2192.4 cm2

5. 2851.8 ft2 11. 27.7 in2

6. 12,080 in2 24. D

7. 2475 in2

8. 1168.5 m2

Finding the area of an equilateral triangle

• The area of any triangle with base length b and height h is given by

A = ½bh. The following formula for equilateral triangles; however, uses ONLY the side length.

* Look at the next slide *

Theorem 11.3 Area of an equilateral triangle • The area of an

equilateral triangle is one fourth the square of the length of the side times

A = ¼ s2

3

3

s s

s

A = ¼ s2 3

Ex. 1: Finding the area of an Equilateral Triangle

• Find the area of an equilateral triangle with 8 inch sides.

A = ¼ s2 3

A = ¼ 82 3

A = ¼ • 64 3

A = • 16 3

A = 16 3


Substitute values.

Simplify.


Simplify.

Using a calculator, the area is about 27.7 square inches.

Ex. 2: Finding the area of an Equilateral Triangle

• Find the area of an equilateral triangle with 12 inch sides.

A = ¼ s2 3

A = ¼ 122 3

A = ¼ • 144 3

A = • 36 3

A = 36 3


Substitute values.

Simplify.


Simplify.

Using a calculator, the area is about 62.4 square inches.

More . . .

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

• You can find the area of any regular n-gon by dividing the polygon into congruent triangles.

a

G

F

E

D C

B

A

H

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

More . . .

A = Area of 1 triangle • # of triangles OR

= ½ • apothem • # of sides • side length OR

= ½ • apothem • perimeter of a polygon

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

a

G

F

E

D C

B

A

H

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

Theorem 11.4 Area of a Regular Polygon • The area of a regular n-gon with side lengths

(s) is half the product of the apothem (a) and the perimeter (P), so

A = ½ aP, or A = ½ a • ns.

NOTE: 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 is n • s, or P = ns

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

Example 3

Find the area of a regular pentagon with an apothem of 4 Feet and side length of 3 Feet.



Apothem = 4 Feet

Numbers of Sides = 5

Side Length = 3

Example 3

Find the area of a regular pentagon with an apothem of 4 Feet and side length of 3 Feet.



Apothem = 4 Feet


Side Length = 3

So

½ • 4 • 3 • 5

= 30 Square Feet

Example 4

Find the area of a regular octagon STOP SIGN with an apothem of 9 Feet and side length of 12 Feet.



Apothem = 9 Feet


Side Length = 12

Example 4

Find the area of a regular octagon STOP SIGN with an apothem of 9 Feet and side length of 12 Feet.



Apothem = 9 Feet


Side Length = 12

So

½ • 9 • 8 • 12

= 432 Square Feet

Ex. 5: Finding the area of a regular dodecagon

• Pendulums. The enclosure on the floor underneath the Foucault Pendulum at the Houston Museum of Natural Sciences in Houston, Texas, is a regular dodecagon with side length of about 4.3 feet and a radius of about 8.3 feet. What is the floor area of the enclosure?

Solution:

• A dodecagon has 12 sides. So, the perimeter of the enclosure is

P = 12(4.3) = 51.6 feet

A B

8.3 ft.

S

Solution:

• In ∆SBT, BT = ½ (BA) = ½ (4.3) = 2.15 feet. Use the Pythagorean Theorem to find the apothem ST.

2.15 ft.

4.3 feet

8.3 feet

T

S

A B

22 15.23.8 a =

a 8 feet

A = ½ aP ½ (8)(51.6) = 206.4 ft. 2

So, the floor area of the enclosure is:

Assignment: 10-3-Test Prep

Pages 548-551 Due Today Tuesday January 15, 2013

39, 40, 44, 45, 46

Exit Quiz – 5 Points

• Find the area of an equilateral triangle with 10 inch sides. PLEASE LEAVE YOUR ANSWER IN RADICAL FORM.

A = ¼ s2 3 Area of an equilateral Triangle

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11.2 Areas of Regular Polygons - Montgomery …...10.3 Areas of Regular Polygons Geometry Mr. Peebles Spring 2013 Bell Ringer • Find the area of an equilateral triangle with 6 inch