9-2: Area and Perimeter of Circles and Regular Polynomials

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