8.5.1 EXAMPLE 1 If , find the area of the 100 sector
shown in Figure 8.49. Use your calculator and round
the answer to the nearest hundredth of a square
inch. Solution becomes A = 100 360 # # 102 L 87.27
in2 A = m 360r 2 mO = 100 90 = 360 1 4 90
120 = 360
on to Theorem 9.2.4 and its applications. Exs. 57
Solution To determine the lateral area, we need the
length of the slant height. [See Figure 9.19(b) on the
preceding page.] The lateral area is . Therefore,
Because the area of the square base is or , the
with radii of lengths R and r, with Prove: Aring = (BC)
2 R 7 r 27 ft2 L 22 7 L 22 7 315 cm 19. Find the area
of a regular hexagon each of whose sides 31. has
length 8 ft. 20. The area of an equilateral triangle is .
If the length of each side of the tria
smaller circle. *23. A circle can be inscribed in the
trapezoid shown. Find the area of that circle. 9 4
cm2 18. Find the exact perimeter and area of the
segment shown, given that and In Exercises 19 and
20, find the exact areas of the shaded regions.
mO
triangle with an inscribed circle. If the sides of the
triangle measure 10 ft, 13 ft, and 13 ft, find the
length of the radius of the inscribed circle. 30. Find a
formula for the area of the shaded region, which
represents one-fourth of an annulus (ring).
The same technique that is used to measure the
volume of the pyramid in Example 5 of Section 9.2
could be used to measure the volumes of the Great
Pyramids. 403 9.1 Prisms, Area, and Volume 9.2
Pyramids, Area, and Volume 9.3 Cylinders and
Cones 9.4 Polyhe
many cubic yards of dirt were removed? 1 yd3 = 27
ft3 e13 29. A cube is a right square prism in which
all edges have the same length. For the cube with
edge e, a) show that the total area is . b) find the
total area if . c) show that the volume is . d) fi
regions in Exercises 27 to 31. 1213 in 10813 in2 29.
30. 27. 28. Square 8 7 7 60 4 6 Two tangent
circles, inscribed in a rectangle Equilateral triangle
10 A C B O a c b 37. Prove that the area of a circle
circumscribed about a square is twice the area of
h(b1 + b2) b c a (a) Figure 8.57 1 b c a a b c A B C 3 2
(b) b 2 = b b 1 = a h = a + b Figure 8.58 b c a a b c I III
II Figure 8.59 Perspective on Application 395 The
total area of regions (triangles) I, II, and III is given
by Equating the areas of the t
has length 12 in., as shown in Figure 8.52. Solution
Let represent the area of the triangle shown.
Because , = (36 - 72) in2 . = 90 360 # # 122 - 1 2 # 12
# 12 Asegment = Asector - A A + Asegment =
Asector A EXAMPLE 5 Find the exact perimeter of
the segme
vertex of the triangle, respectively. Every pyramid
has exactly one base. Square BCDE is the base of the
first pyramid, and GHJ is the base of the second
pyramid. Point A is known as the vertex of the
square pyramid; likewise, point F is the vertex of the
measures 6 in., how long is the belt used in the
pulley system? 60-ft2 L 3.14 100 yd 100 yd 6" 6" 20"
4" 4" 8 cm 60 * Merideth Book/Shutterstock 8.5
More Area Relationships in the Circle 387 Theorem
8.5.1 follows directly from Postulate 23. Sector Area
an
postulate. in. # in. # in. = in 4 # 3 # 2 = 24 /
Corresponding to every solid is a unique positive
number V known as the volume of that solid.
POSTULATE 24 (Volume Postulate) w h Figure 9.10
(a) (b) Figure 9.11 Geometry in the Real World The
frozen solids
lateral area is found by adding the areas of the three
rectangular lateral faces. That is, We use Herons
Formula to find the area of each base. With s = , 1
2(13 + 14 + 15) = 104 in2 + 112 in2 + 120 in2 = 336
in2 L = 8 in. # 13 in. + 8 in. # 14 in. + 8 in
meat or fish. On one occasion, it is said that
Pythagoras came upon a person beating a dog.
Approaching that person, Pythagoras said, Stop
beating the dog, for in this dog lives the soul of my
friend; I recognize him by his voice. In time, the
secrecy, cl
Example 1.) Solution The area of the square base is
or . Because , the formula becomes V = 1 3 (36 in2 )
(4 in.) = 48 in3 V = 1 3Bh 36 in h = 4 in. 2 B = (6 in.)2
s = 6 in. h = 4 in. In Example 5, we apply the
following theorem. This application of the
Py
a Circle Area and Perimeter of Segment Area of
Triangle with Inscribed Circle 396 CHAPTER 8 AREAS
OF POLYGONS AND CIRCLES TABLE 8.3 An Overview
of Chapter 8 Area and Perimeter Relationships
FIGURE DRAWING AREA PERIMETER OR
CIRCUMFERENCE Rectangle A = /w (
is (each side of the base of the pyramid has length s,
and the slant height has length ). The combined
areas of the triangles give the lateral area. Because
there are n triangles, where P is the perimeter of
the base. = 1 2 /P = 1 2 # /(n # s) L = n # 1 2
the prism. Because the lateral edges of this prism
are perpendicular to its base edges, the lateral faces
(like quadrilateral ) are rectangles. Points A, B, C, , ,
and are the vertices of the prism. In Figure 9.1(b),
the lateral edges of the prism are not
Answers are based on Sections 9.1 and 9.2. 1. a) b)
6. In the solid shown, the base is a regular hexagon.
a) Name the vertex of the pyramid. b) Name the
base edges of the pyramid. c) Assuming that lateral
edges are congruent, are the lateral faces also
co
= 1 2r # a + 1 2r # b + 1 2r # c A = A1 + A2 + A3 A = 1
2rP c b a O r (a) c b a 2 1 3 (b) O EXAMPLE 6 Find the
area of a triangle whose sides measure 5 cm, 12 cm,
and 13 cm if the radius of the inscribed circle is 2
cm. See Figure 8.55. Solution With the
shape of a regular hexagonal pyramid. The altitude
of the pyramid has the same length as any side of
the base. If the volume of the interior is 11,972 ft3 ,
find the length of the altitude and of each side of
the base to the nearest foot. 32. The foyer pl
13. Assume that the number of sides in the base of
a pyramid is n. Generalize the results found in
earlier exercises by answering each of the following
questions. a) What is the number of vertices? b)
What is the number of lateral edges? c) What is the
nu
For a regular square pyramid with height 4 in. and
base edges of length 6 in. each, find the length of
the slant height . (See Figure 9.16 on page 415.)
Solution In Figure 9.16, it can be shown that the
apothem to any side has length 3 in. (one-half the
l
BC 25, and AC 17 Find: AABC 5. Given: AB 30, BC 26,
and AC 28 Find: AABC 6. Given: Trapezoid ABCD,
with , BC 6, AD 12, and AB 5 Find: AABCD AB CD ABC
BD DC ABCD AB BD mA = 45 mA = 60 mA =
30 ABCD mC = 90 ABCD 10. Alices mother wants
to wallpaper two adjac
SURFACES AND SOLIDS 15" 13" 14" 8" Figure 9.5
Many students (and teachers) find it easier to
calculate the lateral area of a prism without using
the formula . We illustrate this in Example 2. L = hP
NOTE: The total area of the prism is also known as
its s
total (surface) area of the prism. 9. Suppose that
each of the bases of the hexagonal prism in Exercise
1 has an area of 12 cm2 and that the altitude of the
prism measures 10 cm. Find the volume of the
prism. 10. Suppose that each of the bases of the
tria
Title
Mapping Electric Fields
Abstract
Electric field maps can be produced by mapping an electric fields equipotential lines,
and then connecting them with electric field lines. In this lab this was accomplished for an
electric field consisting of two poi