1. Class Greeting

2. Areas of Regular Polygons and Circles
Chapter 9 – Lesson 2
Areas of Regular Polygons and Circles

3. Objectives
Develop and apply the formulas for the area and circumference of a circle.
Develop and apply the formula for the area of a regular polygon.

4. Vocabulary circle center of a circle center of a regular polygon
apothem
central angle of a regular polygon

5. A circle is the locus of points in a plane that are a fixed distance from a point called the center of the circle.
A circle is named by the symbol  and its center. A has radius r = AB and diameter d = CD.

6. The irrational number  is defined as the ratio of the circumference C to the diameter d, or
Solving for C gives the formula C = d.
Also, since d = 2r, then C = 2r.
The formula for Area of a Circle is A = r2.

7. Example 1A: Finding Measurements of Circles
Find the area of K in terms of .
A = r2
Area of a Circle
A = (3)2
A = 9 in2

8. Example 1B: Finding Measurements of Circles
Find the radius of J if the circumference is (65x + 14) m.
C = 2r
(65x + 14) = 2r
r = (32.5x + 7) m J

9. Example 1C: Finding Measurements of Circles
Find the circumference of M if the area is 25 x2 ft2
Step 1 Use the given area to solve for r.
A = r2
Area of a Circle
25x2 = r2 M
25x2 = r2
5x = r
Step 2 Use the value of r to find the circumference.
C = 2r
Circumference of a Circle
C = 2(5x)
C = 10x ft

10. Check It Out! Example 1
Find the area of A in terms of  in which
C = (4x – 6) m.
A = r2
Area of a Circle
A = (2x – 3)2 m
A = (4x2 – 12x + 9) m2

11. Always wait until the last step to round.
The  key gives the best possible approximation for  on your calculator.
Always wait until the last step to round.
Helpful Hint

12. Example 2: Cooking Application
A pizza-making kit contains three circular baking stones with diameters 24 cm, 36 cm, and 48 cm. Find the area of each stone. Round to the nearest tenth.
24 cm diameter
36 cm diameter
48 cm diameter
A = (12)2
A = (18)2
A = (24)2
≈ cm2
≈ cm2
≈ cm2

13. Check It Out! Example 2
A drum kit contains three drums with diameters of 10 in., 12 in., and 14 in. Find the circumference of each drum to the nearest tenth of an inch.
10 in. diameter in. diameter in. diameter
C = d
C = d
C = d
C = (10)
C = (12)
C = (14)
C ≈ 31.4 in.
C ≈ 37.7 in.
C ≈ 44.0 in.

14. The center of a regular polygon is equidistant from the vertices
The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a side. A central angle of a regular polygon has its vertex at the center, and its sides pass through consecutive vertices. Each central angle measure of a regular n-gon is

15. Definition Apothem [of a regular polygon]
A segment from the center of the figure which is perpendicular to the side is called the apothem
apothem
Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE.

16. Area of a Regular Polygon
Area A of a regular polygon with perimeter P and apothem a is given by:

17. The tangent of an angle in a right triangle is the ratio of the opposite leg length to the adjacent leg length.
Remember!
See page 525.
Soh-Cah-Toa

18. Example 3: Finding the Area of a Regular Polygon
Find the area of regular heptagon with side length 2 ft to the nearest tenth.
Area of a regular polygon
Step 1 Draw the heptagon.
Step 2 Draw an isosceles triangle with its vertex at the center of the heptagon.
Central angle measure is
25.7o
Step 3 Draw a segment that bisects the central angle and the side of the polygon to form a right triangle. 2 a 1 c
This segment is the apothem.
Step 4 Use the tangent ratio to find the apothem.

19. Step 4 Use the tangent ratio to find the apothem.
Example 3 Continued
Step 4 Use the tangent ratio to find the apothem.
Tangent of an angle is
opp. leg
adj. leg a
Step 5 Use the apothem and the given side length to find the area.
Area of a regular polygon
A  14.5 ft2

20. Step 1 Draw the dodecagon.
Check It Out! Example 3
Find the area of a regular dodecagon with side length 5 cm to the nearest tenth.
Step 1 Draw the dodecagon.
Step 2 Draw an isosceles with its vertex at the center of the dodecagon. c
15o
The central angle is 5 a c
2.5
Step 3 Draw a segment that bisects the central angle and the side of the polygon to form a right .
This segment is the apothem.
Step 4 Use the tangent ratio to find the apothem.

21. Check It Out! Example 3 Continued
Step 4 Use the tangent ratio to find the apothem.
Tangent of an angle is
opp. leg
adj. leg
Step 5 Use the apothem and the given side length to find the area.
Area of a regular polygon
A  cm2

22. Triangle
Triangle

23. Kahoot!

24. Lesson Summary: Objectives
Develop and apply the formulas for the area and circumference of a circle.
Develop and apply the formula for the area of a regular polygon.

25. Preview of the Next Lesson:
Objectives
Use the Area Addition Postulate to find the areas of composite figures.
Use composite figures to estimate the areas of irregular shapes.

26. Stand Up Please