Chapter 14

1. Chapter 14 by Julia Duffy and Evan Ribaudo

2. Lesson 1 Vocabulary: Regular polygon- convex polygon that is both equilateral and equiangular Reminder: convex polygon means for each pair of points inside the polygon, the line segment connecting them lies entirely inside the polygon Radius- line segment that connects the center of a regular polygon to a vertex or distance between the center and that vertex Apothem- a perpendicular line segment from a regular polygon’s center to one of its sides.

3. Lesson 1 Definition: *The center of a regular polygon is the center of its circumscribed circle. -Theorem: *Every regular polygon is cyclic Reminder: For a polygon to be cyclic there exists a circle that contains all of its vertices.

4. Lesson 2 Perimeter of a Regular Polygon: Theorem: The perimeter of a regular polygon having n sides is 2Nr, in which N=n sin 180/n and r is its radius. Formula: 2Nr

5. Lesson 3 Area of a Regular Polygon: Theorem: The area of a regular polygon having n sides is Mr2, in which M=n sin 180/n(cos 180/n) and r is its radius. Formula: Mr2 Remember these formulas do not give you exact form!!!

6. Something else to Remember The given formula for the area and perimeter of regular polygons uses the radius, but for some polygons such as a hexagon or square, you can also find the area and perimeter if the apothem or a side is given by dividing the shape into triangles and using what you know about special right triangles (exact form) or trigenometry (rounded form) to find the radius, which you can then use in the given formula.

7. Lesson 4 VOCAB Circumference The circumference of a circle is the limit of the perimeters of inscribed regular polygons Reminder: A regular polygon is a convex polygon that is both equilateral and equiangular

8. Theorems, and Corollaries Theorem 77: If the radius of a circle is r, then its circumference is 2πr. Corollary to Theorem 77: If the diameter of a circle is d, its circumference is πd. Reminder: In circles and regular polygons, 2r=d.

9. Core Concepts Circumference Formula: 2πr or πd. Perimeter Formula of a Regular Polygon: 2Nr in which N=n sin 180/n The equations for perimeter/circumference and area are very similar between circles and regular polygons. The more sides a regular polygon has, the harder it becomes to distinguish from the circle it is inscribed in. As the number of sides a regular polygon increases, so does it’s perimeter, until it reaches it’s limit, which is the perimeter of a circle otherwise known as the circumference.

10. Lesson 5 VOCAB Area: The area of a circle is the limit of areas of the inscribed regular polygons. The relationship between areas of a regular polygon and the circle it is inscribed in are similar to that of their perimeters/circumferences.

11. Theorems and Corollaries Theorem 78: If the radius of a circle is r, its area is πr2.

12. Core Concepts Area Formula of a Circle: πr2 Area Formula of a Regular Polygon: Mr2, in which M=n sin 180/n(cos 180/n) The equations for the area of a regular polygon and a circle are very similar. (Like the equations in lesson 4.) Just as the perimeters of a regular polygon get closer to the circumference of the circle they are inscribed in with an increasing number in size, the polygon’s area also gets closer to the area of the circle it is inscribed in. As the number of sides of the polygon increases, so does the polygon’s area, until it reaches its limit, which is the area of the circle it is inscribed in.

13. Lesson 6 VOCAB Sector: A sector of a circle is a region bounded by an arc of the circle and the two radii to the endpoints of the arc. Area: The area of a sector is (m/360)πr2 where m is the central angle of the sector. Length: The length of a sector’s arc is (m/360)2πr or (m/360)πd.

14. Theorems and Corollaries • Theorem 78.5: If a sector is a certain fraction of a circle, then its area is the same fraction of the circle’s area. If an arc is a certain fraction of a circle, then its length is the same fraction of the circles circumference.

15. Core Concepts • Circles are 360o. When we divide circles into sectors we are able to find individual angle measures, arc lengths, and areas. • For example: If a circular pizza is cut into 8 equal slices, the angle measure of one slice from the center as well as the length of the crust will be 1/8th of the angle measure of the entire pizza. Also, the area of one slice will be 1/8th of the area of the entire pizza.

16. Practice Problems A hexagon has a radius of 6, find the area rounded to the nearest tenth A square has a radius of 4, find the perimeter rounded to the nearest tenth

17. More Practice Problems The circumference of a circle is 12Π. Find the circles radius and area in exact form. A: r=6 and a=36Π If a circle with a radius of 10 is divided into 5 sectors, what is one of the sectors angle measure, and area in exact form? A: \_A=72o a=100Π

18. Summary A regular polygon is cyclic, equilateral and equiangular, containing a radius which connects the center to a vertex of the polygon, and an apothem which is a perpendicular line segment from a regular polygon’s center to one of its sides. The formula for the perimeter of a regular polygon is 2Nr, and the formula for the area of a regular polygon is Mr2.

19. Summary Pt. 2 The circumference of a circle is the limit of the perimeters of inscribed regular polygons. The formula for the circumference of a circle is 2πr. The area of a circle is the limit of areas of the inscribed regular polygons. The formula for the area of a circle is πr2. A sector of a circle is a region bounded by an arc of the circle and the two radii to the endpoints of the arc. The length of a sector’s arc is (m/360)2πr or (m/360)πd. The area of a sector is (m/360)πr2. In both cases, m isis the central angle of the sector.