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Home , Apothem, Dodecagon, Equilateral triangle, Polygon, Regular polygon

Areas of Regular Finding the of an equilateral

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

A = ½bh

The following formula for equilateral ; however, uses ONLY the side length. Area of an

• The area of an equilateral triangle is one fourth the of the length of the side times 3 s s

A = ¼ s2

s

A = ¼ s2 Finding the area of an Equilateral Triangle • Find the area of an equilateral triangle with 8 inch sides. Finding the area of an Equilateral Triangle • Find the area of an equilateral triangle with 8 inch sides.

A = ¼ 3 s2 Area of an equilateral Triangle

A = ¼ 82 Substitute values.

A = ¼ • 64 Simplify.

A = • 16 Multiply ¼ times 64. A = 16 Simplify. Using a calculator, the area is about 27.7 square inches. • The is the F height of a triangle A between the center and two consecutive vertices H of the . a E G B • As in the activity, you can find the area o any

regular n-gon by dividing D C the polygon into congruent triangles. ABCDEF with center G, GA, and apothem GH A = Area of 1 triangle • # of triangles F A

= ( ½ • apothem • side length s) • # H of sides a G B = ½ • apothem • # of sides • side length s E

= ½ • apothem • of a D C polygon Hexagon ABCDEF with center G, This approach can be used to find radius GA, and the area of any . apothem GH Theorem: Area of a Regular Polygon

• The area of a regular n-gon with side (s) is half the product of the apothem (a) and the perimeter (P), so

The number of congruent triangles formed will be A = ½ aP, or A = ½ a • ns. the same as the number of sides of the polygon.

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 • A central of a regular polygon is an angle whose is the center and whose sides contain two consecutive vertices of the polygon. You can divide 360° by the number of sides to find the measure of each central angle of the polygon.

• 360/n = central angle Finding the area of a regular polygon

• A regular is C inscribed in a with 1 radius 1 unit. Find the B area of the pentagon. D 1

A Solution:

• The apply the formula for the area of a regular pentagon, you must B find its apothem and perimeter. • The measure of central ABC is • 360°, or 72°. 1

1 A C 5 D Solution:

• In ∆ABC, the to base AC also bisects B ABC and side AC. The measure 36° of DBC, then is 36°. In ∆BDC, you can use trig ratios to find the lengths of the 1 legs.

C A D One side

• Reminder – rarely in math do you not use something you learned in the past chapters. You will learn and apply after this.

opp adj opp sin = cos = hyp hyp tan = adj B BD cos 36° = 36° AD You have the 1 BD hypotenuse, you know cos 36° = 1 the degrees . . . use cos 36° = BD cosine A D Which one? • Reminder – rarely in math do you not use something you learned in the past chapters. You will learn and apply after this.

opp adj opp sin = cos = hyp hyp tan = adj B DC 36° sin 36° = BC You have the 1 1 DC hypotenuse, you know sin 36° = 1 the degrees . . . use sin 36° = DC D C SO . . .

• So the pentagon has an apothem of a = BD = cos 36° and a perimeter of P = 5(AC) = 5(2 • DC) = 10 sin 36°. Therefore, the area of the pentagon is

A = ½ aP = ½ (cos 36°)(10 sin 36°)  2.38 square units. Finding the area of a regular

• 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 S

8.3 ft.

A B 4.3 ft. Solution: S

• In ∆SBT, BT = ½ (BA) = ½ (4.3) = 2.15 feet. Use the to find the apothem ST. 8.3 feet

2.15 ft. A B T 4.3 feet a = 8.32  2.152 a  8 feet So, the floor area of the enclosure is: A = ½ aP  ½ (8)(51.6) = 206.4 ft. 2