Lesson 11-4 Areas of Regular Polygons (page 440) Essential Question How can you calculate the area of any figure?

Areas of Regular Polygons

Student Activity Using a compass, construct a circle with a given radius.

r

Mark off congruent arcs with chords equal to the radius of the circle. Start where the radius meets the circle.

r

Connect the consecutive points on the circle to form a polygon.

r r r r r r r

r r r r r r r The figure formed is a regular hexagon.

A regular polygon is both equilateral and equiangular.

Any regular polygon can be inscribed in a circle.

Many of the terms associated with circles are also used with regular polygons.

The center of a regular polygon is the center of the circumscribed circle.

The radius of the regular polygon is the distance from the center to a vertex. r

A central angle is an angle formed by two radii drawn to two consecutive vertices.

The measure of a central angle of a regular polygon with “n” sides is _____.

The apothem of a regular polygon is the distance from the center to a side.

r a Check out the apothem and radius.

r a Check out the apothem and radius.

r a Check out the apothem and radius.

r a Check out the apothem and radius.

r a Check out the apothem and radius.

The area of a regular polygon is equal to half the product of the apothem and the perimeter. Theorem 11-6 A regular polygon = ½ ap

Here is how that works … Take a regular n-gon and draw all its radii. That will produce “n” triangles. The area of one of those triangles is …

r s s s s s s r a

Take a regular n-gon and draw all its radii. That will produce “n” triangles. The area of one of those triangles is … Take this times the “n” triangles …

A regular triangle = ½ ap r a ss Also known as (a.k.a.): Equilateral Triangle Equiangular Triangle ½ s

Don’t miss this! r a ss 120º 60º 30º 60º 30º See the 30º-60º-90º ∆’s! 2a = ½ s 30º

A regular quadrilateral = ½ ap r a s s s a.k.a.: square ½ s

r a s s s Don’t miss this! ½ s 45º 90º 45º See the 45º-45º-90º ∆’s! a = = a 2a = = 2a 45º

A regular hexagon = ½ ap r a s s ss s ½ s

r a s s ss s Don’t miss this! ½ s 120º 60º 120º 60º 30º 60º

r a s s ss s Don’t miss this! ½ s 60º See the 30º-60º-90º ∆’s! 60º 30º 60º s = r = = r ½ r =

Example #1 Find the perimeter and area of a regular triangle with its apothem equal to 9 feet. 9’9’ 30º 60º

Example #2 Find the apothem and radius of a regular quadrilateral with an area of 100 sq. yd. 10 yd 5 yd 45º a = 5 yda r

Example #3 Find the area of a regular hexagon with a side equal to 12 units. 12u 6u 60º 30º

Assignment Written Exercises on pages 443 & 444 REQUIRED: 1 to 15 odd numbers * Bonus: 18, 19, 20 * How can you calculate the area of any figure? UPDATE YOUR STUDENT AID CARD!