1. Circles and Regular Polygons
Developing Formulas
Circles and Regular Polygons
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
Holt Geometry

2. Warm Up Find the unknown side lengths in each special right triangle.
1. a 30°-60°-90° triangle with hypotenuse 2 ft
2. a 45°-45°-90° triangle with leg length 4 in.
3. a 30°-60°-90° triangle with longer leg length 3m

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.
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 d = 2r, so C = 2r.

6. You can use the circumference of a circle to find its area
You can use the circumference of a circle to find its area. Divide the circle and rearrange the pieces to make a shape that resembles a parallelogram.
The base of the parallelogram is about half the circumference, or r, and the height is close to the radius r. So A   r · r =  r2.
The more pieces you divide the circle into, the more accurate the estimate will be.

8. Example 1A: Finding Measurements of Circles
Find the area of K in terms of .
A = r2
Area of a circle.
Divide the diameter by 2 to find the radius, 3.
A = (3)2
A = 9 in2
Simplify.

9. Example 1B: Finding Measurements of Circles
Find the radius of J if the circumference is (65x + 14) m.
C = 2r
Circumference of a circle
(65x + 14) = 2r
Substitute (65x + 14) for C.
r = (32.5x + 7) m
Divide both sides by 2.

10. 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
Substitute 25x2 for A.
25x2 = r2
Divide both sides by .
Take the square root of both sides.
5x = r

11. Example 1C Continued
Step 2 Use the value of r to find the circumference.
C = 2r
C = 2(5x)
Substitute 5x for r.
C = 10x ft
Simplify.

12. 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.
Divide the diameter by 2 to find the radius, 2x – 3.
A = (2x – 3)2 m
A = (4x2 – 12x + 9) m2
Simplify.

13. 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

14. 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

15. 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.
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.

16. 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

17. Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE.

18. To find the area of a regular n-gon with side length s and apothem a, divide it into n congruent isosceles triangles.
area of each triangle:
total area of the polygon:
The perimeter is P = ns.

20. Example 3A: Finding the Area of a Regular Polygon
Find the area of regular heptagon with side length 2 ft to the nearest tenth.
Step 1 Draw the heptagon. Draw an isosceles triangle with its vertex at the center of the heptagon. The central angle is .
Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.

21. Example 3A Continued
Step 2 Use the tangent ratio to find the apothem.
The tangent of an angle is
opp. leg
adj. leg
Solve for a.

22. Example 3A Continued
Step 3 Use the apothem and the given side length to find the area.
Area of a regular polygon
The perimeter is 2(7) = 14ft.
Simplify. Round to the nearest tenth.
A  14.5 ft2

23. The tangent of an angle in a right triangle is the ratio of the opposite leg length to the adjacent leg length. See page 525.
Remember!

24. Example 3B: Finding the Area of a Regular Polygon
Find the area of a regular dodecagon with side length 5 cm to the nearest tenth.
Step 1 Draw the dodecagon. Draw an isosceles triangle with its vertex at the center of the
dodecagon. The central angle is
Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.

25. Example 3B Continued
Step 2 Use the tangent ratio to find the apothem.
The tangent of an angle is
opp. leg
adj. leg
Solve for a.

26. Example 3B Continued
Step 3 Use the apothem and the given side length to find the area.
Area of a regular polygon
The perimeter is 5(12) = 60 ft.
Simplify. Round to the nearest tenth.
A  cm2

27. Check It Out! Example 3
Find the area of a regular octagon with a side length of 4 cm.
Step 1 Draw the octagon. Draw an isosceles triangle with its vertex at the center of the octagon. The
central angle is
Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.

28. Check It Out! Example 3 Continued
Step 2 Use the tangent ratio to find the apothem
The tangent of an angle is
opp. leg
adj. leg
Solve for a.

29. Check It Out! Example 3 Continued
Step 3 Use the apothem and the given side length to find the area.
Area of a regular polygon
The perimeter is 4(8) = 32cm.
Simplify. Round to the nearest tenth.
A ≈ 77.3 cm2

30. Lesson Quiz: Part I
Find each measurement.
1. the area of D in terms of 
A = 49 ft2
2. the circumference of T in which A = 16 mm2
C = 8 mm

31. Lesson Quiz: Part II
Find each measurement.
3. Speakers come in diameters of 4 in., 9 in., and 16 in. Find the area of each speaker to the nearest tenth.
A1 ≈ 12.6 in2 ; A2 ≈ 63.6 in2 ; A3 ≈ in2
Find the area of each regular polygon to the nearest tenth.
4. a regular nonagon with side length 8 cm
A ≈ cm2
5. a regular octagon with side length 9 ft
A ≈ ft2