The document discusses formulas for circles and regular polygons. It defines key terms like radius, diameter, circumference, area, apothem, and central angle. Examples are provided for calculating circle measurements like circumference and area using formulas like C=2πr and A=πr^2. A formula is developed for the area of a regular polygon by dividing it into congruent triangles: A=(1/2)aps, where a is the apothem, p is the perimeter, and s is the side length. Worked examples demonstrate calculating measurements for circles and finding areas of regular polygons.

1. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons9-2
Developing Formulas for
Circles and Regular Polygons
Holt Geometry
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz

2. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
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. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
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.
Objectives

4. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
circle
center of a circle
center of a regular polygon
apothem
central angle of a regular polygon
Vocabulary

5. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
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.
Solving for C gives the formula
C = πd. Also d = 2r, so C = 2πr.
The irrational number π
is defined as the ratio of
the circumference C to
the diameter d, or

6. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
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 πr, and the
height is close to the radius r.
So A ≅ π r · r = π r2
.
The more pieces you divide
the circle into, the more
accurate the estimate will
be.

7. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons

8. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
Find the area of K in terms of π.
Example 1A: Finding Measurements of Circles
A = πr2
Area of a circle.
Divide the diameter by 2
to find the radius, 3.
Simplify.
A = π(3)2
A = 9π in2

9. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
Find the radius of J if the circumference is
(65x + 14)π m.
Example 1B: Finding Measurements of Circles
Circumference of a circle
Substitute (65x + 14)π for C.
Divide both sides by 2π.
C = 2πr
(65x + 14)π = 2πr
r = (32.5x + 7) m

10. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
Find the circumference of M if the area is
25 x2
π ft2
Example 1C: Finding Measurements of Circles
Step 1 Use the given area to solve for r.
Area of a circle
Substitute 25x2
π for A.
Divide both sides by π.
Take the square root of
both sides.
A = πr2
25x2
π = πr2
25x2
= r2
5x = r

11. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
Example 1C Continued
Step 2 Use the value of r to find the circumference.
Substitute 5x for r.
Simplify.
C = 2π(5x)
C = 10xπ ft
C = 2πr

12. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
Check It Out! Example 1
Find the area of A in terms of π in which
C = (4x – 6)π m.
A = πr2 Area of a circle.
A = π(2x – 3)2
m
A = (4x2
– 12x + 9)π m2
Divide the diameter by 2
to find the radius, 2x – 3.
Simplify.

13. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
The π key gives the best possible
approximation for π on your calculator.
Always wait until the last step to round.
Helpful Hint

14. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
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.
Example 2: Cooking Application
24 cm diameter 36 cm diameter 48 cm diameter
A = π(12)2
A = π(18)2
A = π(24)2
≈ 452.4 cm2
≈ 1017.9 cm2
≈ 1809.6 cm2

15. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
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 12 in. diameter 14 in. diameter
C = πd C = πd C = πd
C = π(10) C = π(12) C = π(14)
C = 31.4 in. C = 37.7 in. C = 44.0 in.

16. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
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. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
Regular pentagon DEFGH has a center C,
apothem BC, and central angle ∠DCE.

18. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
To find the area of a regular n-gon with side
length s and apothem a, divide it into n
congruent isosceles triangles.
The perimeter is P = ns.
area of each triangle:
total area of the polygon:

19. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons

20. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
Find the area of regular heptagon with side
length 2 ft to the nearest tenth.
Example 3A: Finding the Area of a Regular Polygon
Draw a segment that bisects the central angle and
the side of the polygon to form a right triangle.
Step 1 Draw the heptagon. Draw an isosceles
triangle with its vertex at the center of the
heptagon. The central angle is °.

21. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
Example 3A Continued
tan 25.7°( )=
1
a
Solve for a.
The tangent of an angle is .
opp. leg
adj. leg
Step 2 Use the tangent ratio to find the apothem.

22. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
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. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
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. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
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.
Draw a segment that bisects the central angle and
the side of the polygon to form a right triangle.
Step 1 Draw the dodecagon. Draw an isosceles
triangle with its vertex at the center of the
dodecagon. The central angle is .

25. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
Example 3B Continued
Solve for a.
The tangent of an angle is .
opp. leg
adj. leg
Step 2 Use the tangent ratio to find the apothem.

26. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
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 ≈ 279.9 cm2

27. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
Check It Out! Example 3
Find the area of a regular octagon with a side
length of 4 cm.
Draw a segment that bisects the central angle and
the side of the polygon to form a right triangle.
Step 1 Draw the octagon. Draw an isosceles triangle
with its vertex at the center of the octagon. The
central angle is .

28. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
Step 2 Use the tangent ratio to find the apothem
Solve for a.
Check It Out! Example 3 Continued
The tangent of an angle is .
opp. leg
adj. leg

29. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
Step 3 Use the apothem and the given side length
to find the area.
Check It Out! Example 3 Continued
Area of a regular polygon
The perimeter is 4(8) = 32cm.
Simplify. Round to the nearest
tenth.
A ≈ 77.3 cm2

30. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
Lesson Quiz: Part I
Find each measurement.
1. the area of D in terms of π
A = 49π ft2
2. the circumference of T in which A = 16π mm2
C = 8π mm

31. Holt Geometry
9-2
Developing Formulas for
Circles and Regular Polygons
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 ≈ 201.1 in2
Find the area of each regular polygon to the
nearest tenth.
4. a regular nonagon with side length 8 cm
A ≈ 395.6 cm2
5. a regular octagon with side length 9 ft
A ≈ 391.1 ft2