NAME ______________________________________________ DATE
____________ PERIOD _____
11-1 Study Guide and Intervention
Areas of Parallelograms
Areas of Parallelograms
A parallelogram is a quadrilateral with both pairs of
opposite sides parallel. Any side of a parallelogram can be called a base. Each base has a
corresponding altitude, and the length of the altitude is the height of the parallelogram.
The area of a parallelogram is the product of the base and the height.
A
B
h
D
C
Tb
Lesson 11-1
If a parallelogram has an area of A square units,
a base of b units, and a height of h units,
then A bh.
Area of a Parallelogram
The area of parallelogram
ABCD is CD AT.
Example
Find the area of parallelogram EFGH.
E
F
A bh
Area of a parallelogram
30(18) b 30, h 18
540
Multiply.
The area is 540 square meters.
18 m
H
30 m
G
Exercises
Find the area of each parallelogram.
1.
2.
3.
1.6 cm
16 ft
60
1.6 cm
24 in.
18 ft
Find the area of each shaded region.
4. WXYZ and ABCD are
rectangles.
32 cm
W
A
16 cm
5. All angles are right
angles.
3 ft
X
E
3 ft
B
8 ft
5 cm
3 ft
18 in.
3 ft
D 12 cm C
Z
6. EFGH and NOPQ are
rectangles; JKLM is a
square.
6 ft
Y
12 ft
2 ft
H
J
KN
O
F
9 in.
9 in.
M 9 in. L Q 12 in. P
30 in.
G
7. The area of a parallelogram is 3.36 square feet. The base is 2.8 feet. If the measures of
the base and height are each doubled, find the area of the resulting parallelogram.
8. A rectangle is 4 meters longer than it is wide. The area of the rectangle is 252 square
meters. Find the length.
©
Glencoe/McGraw-Hill
611
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
11-1 Study Guide and Intervention
(continued)
Areas of Parallelograms
Parallelograms on the Coordinate Plane
To find the area of a quadrilateral on
the coordinate plane, use the Slope Formula, the Distance Formula, and properties of
parallelograms, rectangles, squares, and rhombi.
Example
The vertices of a quadrilateral are A(2, 2),
B(4, 2), C(5, 1), and D(1, 1).
a. Determine whether the quadrilateral is a square,
a rectangle, or a parallelogram.
Graph the quadrilateral. Then determine the slope of each side.
y
B
A
O
D
x
C
22
4 (2)
1 (1)
slope of C
D
or 0
1 5
2 (1)
slope A
D
or 3
2 (1)
1 2
slope B
C
or 3
54
B
or 0
slope of A
Opposite sides have the same slope. The slopes of consecutive sides are not negative
reciprocals of each other, so consecutive sides are not perpendicular. ABCD is a
parallelogram; it is not a rectangle or a square.
b. Find the area of ABCD.
From the graph, the height of the parallelogram is 3 units and AB |4 (2)| 6.
A bh
Area of a parallelogram
6(3)
b 6, h 3
2
18 units
Multiply.
Exercises
Given the coordinates of the vertices of a quadrilateral, determine whether the
quadrilateral is a square, a rectangle, or a parallelogram. Then find the area.
1. A(1, 2), B(3, 2), C(3, 2), and D(1, 2)
2. R(1, 2), S(5, 0), T(4, 3), and U(2, 1)
3. C(2, 3), D(3, 3), E(5, 0), and F(0, 0)
4. A(2, 2), B(0, 2), C(4, 0), and D(2, 4)
5. M(2, 3), N(4, 1), P(2, 1), and R(4, 3)
6. D(2, 1), E(2, 4), F(1, 4), and G(1, 1)
©
Glencoe/McGraw-Hill
612
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
11-1 Skills Practice
Area of Parallelograms
Find the perimeter and area of each parallelogram. Round to the nearest tenth if
necessary.
1.
2.
30 cm
5.5 ft
60
4 ft
20 cm
3.
4.
14 yd
26 in.
22 in.
7 yd
5.
Lesson 11-1
60
45
45
6.
3.4 m
18.5 km
9 km
Find the area of each figure.
7.
8.
6
1
8
2 3
1
2
2
1
1
1
6
2
2
2
2 3
2
2
1
2 4
2
6
COORDINATE GEOMETRY Given the coordinates of the vertices of a quadrilateral,
determine whether it is a square, a rectangle, or a parallelogram. Then find the
area of the quadrilateral.
9. A(4, 2), B(1, 2), C(1, 1),
D(4, 1)
11. D(5, 1), E(7, 1), F(4, 4),
G(8, 4)
©
Glencoe/McGraw-Hill
10. P(3, 3), Q(1, 3), R(1, 3),
S(3, 3)
12. R(2, 3), S(4, 10), T(12, 10),
U(10, 3)
613
Glencoe Geometry
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11-2 Study Guide and Intervention
Areas of Triangles, Trapezoids, and Rhombi
Areas of Triangles
The area of a triangle is half the area of a rectangle with the same
base and height as the triangle.
If a triangle has an area of A square units, a base of b units,
X
1
2
and a corresponding height of h units, then A bh.
h
Z
Example
Y
b
Find the area of the triangle.
1
A bh
2
1
(24)(28)
2
28 m
Area of a triangle
b 24, h 28
24 m
336
Multiply.
Lesson 11-2
The area is 336 square meters.
Exercises
Find the area of each figure.
1.
2.
21
20
20
26
16
56
33
3.
20
4.
60
10
24
5.
6.
18
21 54
24
15
60
24
10
7. The area of a triangle is 72 square inches. If the height is 8 inches, find the length of
the base.
8. A right triangle has a perimeter of 36 meters, a hypotenuse of 15 meters, and a leg of
9 meters. Find the area of the triangle.
©
Glencoe/McGraw-Hill
617
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
11-2 Study Guide and Intervention
(continued)
Areas of Triangles, Trapezoids, and Rhombi
Areas of Trapezoids and Rhombi
The area of a trapezoid is the product of half the
height and the sum of the lengths of the bases. The area of a rhombus is half the product of
the diagonals.
If a trapezoid has an area of A square units, bases of
b1 and b2 units, and a height of h units, then
If a rhombus has an area of A square units and
diagonals of d1 and d2 units, then
A h(b1 b2).
A d1d2.
1
2
1
2
b1
d2
h
d1
b2
Example
Find the area of the trapezoid.
1
A h(b1 b2)
2
1
(15)(18 40)
2
435
18 m
Area of a trapezoid
15 m
h 15, b1 18, b2 40
40 m
Simplify.
The area is 435 square meters.
Exercises
Find the area of each quadrilateral given the coordinates of the vertices.
1.
2.
10
20
20
12
10
3.
60
4.
32
32
16
16
18
5.
13
6.
28
13
13
12
13
24
7. The area of a trapezoid is 144 square inches. If the height is 12 inches, find the length of
the median.
8. A rhombus has a perimeter of 80 meters and the length of one diagonal is 24 meters.
Find the area of the rhombus.
©
Glencoe/McGraw-Hill
618
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
11-2 Skills Practice
Areas of Triangles, Trapezoids, and Rhombi
Find the area of each figure. Round to the nearest tenth if necessary.
1.
2.
4 ft
3.
28 cm
22 in.
3.5 ft
34 cm
6 ft
25 in.
12 in.
21 cm
40 cm
Find the area of each quadrilateral given the coordinates of the vertices.
5. rhombus HIJK
H(4, 3), I(2, 7), J(0, 3), K(2, 1)
Lesson 11-2
4. trapezoid WXYZ
W(5, 3), X(3, 3), Y(6, 3), Z(8, 3)
Find the missing measure for each figure.
6. Trapezoid RSTU has an area of
935 square centimeters. Find the
height of RSTU.
S
30 cm
7. Trapezoid JKLM has an area of
7.5 square inches. Find ML.
T
5 in.
J
K
2 in.
R
U
55 cm
8. Triangle ABC has an area of
1050 square meters. Find the
height of ABC.
M
9. Rhombus EFGH has an area of
750 square feet. If EG is 50 feet,
find FH.
F
B
A
©
L
60 m
Glencoe/McGraw-Hill
C
E
619
G
H
Glencoe Geometry
NAME ______________________________________________ DATE
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11-3 Study Guide and Intervention
Areas of Regular Polygons and Circles
Areas of Regular Polygons
In a regular polygon, the segment
drawn from the center of the polygon perpendicular to the opposite side
is called the apothem. In the figure at the right, A
P
is the apothem and
R
A
is the radius of the circumscribed circle.
U
V
T
A
P
Area of a Regular Polygon
R
If a regular polygon has an area of A square units,
a perimeter of P units, and an apothem of a units,
S
1
2
then A Pa.
Example 1
Example 2
Verify the formula
1
A Pa for the regular pentagon above.
2
For RAS, the area is
1
2
1
2
A bh (RS)(AP). So the area of the
12 pentagon is A 5 (RS)(AP). Substituting
P for 5RS and substituting a for AP, then
Find the area of regular
pentagon RSTUV above if its perimeter
is 60 centimeters.
First find the apothem.
360
The measure of central angle RAS is or
5
72. Therefore mRAP 36. The perimeter
is 60, so RS 12 and RP 6.
RP
AP
6
tan 36° AP
6
AP tan 36°
tan RAP 1
2
A Pa.
The area is about 248 square centimeters.
Exercises
Find the area of each regular polygon. Round to the nearest tenth.
1.
2.
14 m
3.
15 in.
10 in.
4.
5.
6.
5
3 cm
10.9 m
7.5 m
10 in.
©
Glencoe/McGraw-Hill
623
Glencoe Geometry
Lesson 11-3
8.26
1
1
So A Pa 60(8.26) or 247.7.
2
2
NAME ______________________________________________ DATE
____________ PERIOD _____
11-3 Study Guide and Intervention
(continued)
Areas of Regular Polygons and Circles
Areas of Circles
As the number of sides of a regular polygon increases, the polygon gets
closer and closer to a circle and the area of the polygon gets closer to the area of a circle.
Area of a Circle
If a circle has an area of A square units
and a radius of r units, then A r 2.
O
r
Example
Circle Q is inscribed in square RSTU. Find the
area of the shaded region.
A side of the square is 40 meters, so the radius of the circle is 20 meters.
R
The shaded area is
Area of RSTU Area of circle Q
402 r2
1600 400
1600 1256.6
343.4 m2
U
S
Q
40 m
40 m
T
Exercises
Find the area of each shaded region. Assume that all polygons are regular. Round
to the nearest tenth.
1.
2.
3.
2 in. 2 in.
16 m
4m
4.
5.
6.
4 in.
12 m
12 m
©
Glencoe/McGraw-Hill
624
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
11-3 Skills Practice
Areas of Regular Polygons and Circles
Find the area of each regular polygon. Round to the nearest tenth.
1. a pentagon with a perimeter of 45 feet
2. a hexagon with a side length of 4 inches
3. a nonagon with a side length of 8 meters
4. a triangle with a perimeter of 54 centimeters
Find the area of each circle. Round to the nearest tenth.
5. a circle with a radius of 6 yards
Find the area of each shaded region. Assume that all polygons are regular. Round
to the nearest tenth.
7.
8.
4m
3 in.
8m
9.
10.
5 cm
4 ft
©
Glencoe/McGraw-Hill
625
Glencoe Geometry
Lesson 11-3
6. a circle with a diameter of 18 millimeters
NAME ______________________________________________ DATE
____________ PERIOD _____
11-3 Practice
Areas of Regular Polygons and Circles
Find the area of each regular polygon. Round to the nearest tenth.
1. a nonagon with a perimeter of 117 millimeters
2. an octagon with a perimeter of 96 yards
Find the area of each circle. Round to the nearest tenth.
3. a circle with a diameter of 26 feet
4. a circle with a circumference of 88 kilometers
Find the area of each shaded region. Assume that all polygons are regular. Round
to the nearest tenth.
5.
6.
12 cm
4.4 in.
7.
8.
9m
25 ft
DISPLAYS For Exercises 9 and 10, use the following information.
A display case in a jewelry store has a base in the shape of a regular octagon. The length of
each side of the base is 10 inches. The owners of the store plan to cover the base in black
velvet.
9. Find the area of the base of the display case.
10. Find the number of square yards of fabric needed to cover the base.
©
Glencoe/McGraw-Hill
626
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
12-1 Study Guide and Intervention
(continued)
Three-Dimensional Figures
Identify Three-Dimensional Figures
A polyhedron is a solid with all flat
surfaces. Each surface of a polyhedron is called a face, and each line segment where faces
intersect is called an edge. Two special kinds of polyhedra are prisms, for which two faces
are congruent, parallel bases, and pyramids, for which one face is a base and all the other
faces meet at a point called the vertex. Prisms and pyramids are named for the shape of
their bases, and a regular polyhedron has a regular polygon as its base.
pentagonal
prism
square
pyramid
pentagonal
pyramid
rectangular
prism
cylinder
cone
sphere
Other solids are a cylinder, which has congruent circular bases in parallel planes, a cone,
which has one circular base and a vertex, and a sphere.
Example
a.
Identify each solid. Name the bases, faces, edges, and vertices.
b.
E
D
O
C
A
P
B
The figure is a rectangular pyramid. The base is
rectangle ABCD, and the four faces ABE, BCE,
CDE, and ADE meet at vertex E. The edges are
B
A
, B
C
, C
D
, A
D
, A
E
, B
E
, C
E
, and D
E
. The vertices
are A, B, C, D, and E.
This solid is a cylinder. The
two bases are O and P.
Exercises
Identify each solid. Name the bases, faces, edges, and vertices.
1.
2.
R
T
S
3.
Z
4.
U
Y
X
S
R
Glencoe/McGraw-Hill
E
D
Q
P W
©
F
C
V
A
662
B
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
12-1 Skills Practice
Three-Dimensional Figures
Draw the back view and corner view of a figure given each orthogonal drawing.
2.
top view
left view
back view
front view
right view
top view
corner view
back view
left view
front view
right view
Lesson 12-1
1.
corner view
Identify each solid. Name the bases, faces, edges, and vertices.
3.
Y X
R
S
V
U
W
T
4.
F
A
B
C
D
E
5.
R
S
©
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663
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
12-3 Study Guide and Intervention
Surface Areas of Prisms
Lateral Areas of Prisms
Here are some characteristics
altitude
of prisms.
•
•
•
•
The bases are parallel and congruent.
The lateral faces are the faces that are not bases.
The lateral faces intersect at lateral edges, which are parallel.
The altitude of a prism is a segment that is perpendicular
to the bases with an endpoint in each base.
• For a right prism, the lateral edges are perpendicular to the
bases. Otherwise, the prism is oblique.
Lateral Area
of a Prism
lateral
edge
lateral
face
pentagonal prism
If a prism has a lateral area of L square units, a height of h units,
and each base has a perimeter of P units, then L Ph.
Example
Find the lateral area of the regular pentagonal prism above if each
base has a perimeter of 75 centimeters and the altitude is 10 centimeters.
L Ph
Lateral area of a prism
75(10)
P 75, h 10
750
Multiply.
The lateral area is 750 square centimeters.
Exercises
1.
Lesson 12-3
Find the lateral area of each prism.
2.
3m
10 in.
10 m
8 in.
4m
15 in.
3.
4.
10 cm
10 cm
9 cm
6 in.
12 cm
18 in.
20 cm
5.
6.
4 in.
12 in.
16 m
4 in.
©
Glencoe/McGraw-Hill
4m
673
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
12-3 Study Guide and Intervention
(continued)
Surface Areas of Prisms
Surface Areas of Prisms
The surface area of a prism is the
lateral area of the prism plus the areas of the bases.
6 cm
6 cm
h
If the total surface area of a prism is T square units, its height is
h units, and each base has an area of B square units and a
perimeter of P units, then T L 2B.
Surface Area
of a Prism
6 cm
10 cm
Example
Find the surface area of the triangular prism above.
Find the lateral area of the prism.
L Ph
(18)(10)
180 cm2
Lateral area of a prism
P 18, h 10
Multiply.
Find the area of each base. Use the Pythagorean Theorem to find the height of the
triangular base.
h2 32 62
h2 27
h 3
3
Pythagorean Theorem
Simplify.
Take the square root of each side.
1
B base height
Area of a triangle
2
1
(6)(33
) or 15.6 cm2
2
The total area is the lateral area plus the area of the two bases.
T 180 2(15.6)
211.2 cm2
Substitution
Simplify.
Exercises
Find the surface area of each prism. Round to the nearest tenth if necessary.
1.
2.
10 in.
3.
5m
4m
8 in.
3m
24 in.
4.
5.
12 m
8 in.
8 in.
6 in.
©
12 in.
Glencoe/McGraw-Hill
6m
6m
6.
8m
15 in.
8 in.
8m
8m
674
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
12-3 Skills Practice
Surface Areas of Prisms
Find the lateral area of each prism.
1.
2.
6
12
12
10
8
12
3.
4.
9
6
8
9
12
5
10
Find the surface area of each prism. Round to the nearest tenth if necessary.
5.
6.
7
8
13
6
15
18
7.
8.
4
6
10
3
7
9
©
Glencoe/McGraw-Hill
675
Glencoe Geometry
Lesson 12-3
9
NAME ______________________________________________ DATE
____________ PERIOD _____
12-4 Study Guide and Intervention
Surface Areas of Cylinders
Lateral Areas of Cylinders
A cylinder is a solid whose
bases are congruent circles that lie in parallel planes. The axis
of a cylinder is the segment whose endpoints are the centers of
these circles. For a right cylinder, the axis and the altitude of
the cylinder are equal. The lateral area of a right cylinder is the
circumference of the cylinder multiplied by the height.
Lateral Area
of a Cylinder
base
axis
base
height
radius of base
If a cylinder has a lateral area of L square units, a height of h units,
and the bases have radii of r units, then L 2rh.
Example
Find the lateral area of the cylinder above if the radius of the base
is 6 centimeters and the height is 14 centimeters.
L 2rh
Lateral area of a cylinder
2(6)(14)
Substitution
527.8
Simplify.
The lateral area is about 527.8 square centimeters.
Exercises
Find the lateral area of each cylinder. Round to the nearest tenth.
1.
2.
4 cm
10 in.
6 in.
12 cm
3.
4.
3 cm
8 cm
6 cm
5.
Lesson 12-4
3 cm
20 cm
6.
2m
1m
12 m
4m
©
Glencoe/McGraw-Hill
679
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
12-4 Study Guide and Intervention
(continued)
Surface Areas of Cylinders
Surface Areas of Cylinders
The surface area of a cylinder
is the lateral area of the cylinder plus the areas of the bases.
base
lateral area
Surface Area
of a Cylinder
If a cylinder has a surface area of T square units, a height of
h units, and the bases have radii of r units, then T 2rh 2r 2.
base
Example
Find the surface area of the cylinder.
Find the lateral area of the cylinder. If the diameter is
12 centimeters, then the radius is 6 centimeters.
L Ph
(2r)h
2(6)(14)
527.8
12 cm
14 cm
Lateral area of a cylinder
P 2r
r 6, h 14
Simplify.
Find the area of each base.
B r2
(6)2
113.1
Area of a circle
r6
Simplify.
The total area is the lateral area plus the area of the two bases.
T 527.8 113.1 113.1 or 754 square centimeters.
Exercises
Find the surface area of each cylinder. Round to the nearest tenth.
1.
2.
2m
10 in.
2m
12 in.
3.
4.
3 yd
8 in.
12 in.
2 yd
5.
6.
2m
©
8 in.
15 m
Glencoe/McGraw-Hill
20 in.
680
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
12-4 Skills Practice
Surface Areas of Cylinders
Find the surface area of a cylinder with the given dimensions. Round to the
nearest tenth.
1. r 10 in., h 12 in.
2. r 8 cm, h 15 cm
3. r 5 ft, h 20 ft
4. d 20 yd, h 5 yd
5. d 8 m, h 7 m
6. d 24 mm, h 20 mm
Find the surface area of each cylinder. Round to the nearest tenth.
7.
5 ft
8.
4m
7 ft
8.5 m
Find the radius of the base of each cylinder.
10. The surface area is 100.5 square inches, and the height is 6 inches.
11. The surface area is 226.2 square centimeters, and the height is 5 centimeters.
12. The surface area is 1520.5 square yards, and the height is 14.2 yards.
©
Glencoe/McGraw-Hill
681
Glencoe Geometry
Lesson 12-4
9. The surface area is 603.2 square meters, and the height is 10 meters.
NAME ______________________________________________ DATE
____________ PERIOD _____
12-5 Study Guide and Intervention
Surface Areas of Pyramids
Lateral Areas of Regular Pyramids
Here are some properties of pyramids.
• The base is a polygon.
• All of the faces, except the base, intersect in a common point known as the vertex.
• The faces that intersect at the vertex, which are called lateral faces, are triangles.
For a regular pyramid, the base is a regular polygon and the slant height is the
height of each lateral face.
If a regular pyramid has a lateral area of L square units, a slant height of units,
1
and its base has a perimeter of P units, then L 2P.
Lateral Area of a
Regular Pyramid
Example
The roof of a barn is a regular octagonal
pyramid. The base of the pyramid has sides of 12 feet,
and the slant height of the roof is 15 feet. Find the
lateral area of the roof.
The perimeter of the base is 8(12) or 96 feet.
1
2
1
(96)(15)
2
L P
vertex
lateral face
slant height
lateral edge
base
Lateral area of a pyramid
P 96, 15
720
Multiply.
The lateral area is 720 square feet.
Exercises
Find the lateral area of each regular pyramid. Round to the nearest tenth if
necessary.
1.
2.
8 cm
15 cm
8 cm
10 ft
3.5 ft
8 cm
3.
4.
42 m
5.
6.
18 in.
©
6 ft
Lesson 12-5
60
20 m
45 12 yd
60
Glencoe/McGraw-Hill
685
Glencoe Geometry
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____________ PERIOD _____
12-5 Study Guide and Intervention
(continued)
Surface Areas of Pyramids
Surface Areas of Regular Pyramids
The surface area
of a regular pyramid is the lateral area plus the area of the base.
lateral edge
slant height
Surface Area of a
Regular Pyramid
If a regular pyramid has a surface area of T square units,
a slant height of units, and its base has a perimeter of
1
P units and an area of B square units, then T P B.
height
base
2
Example
For the regular square pyramid above, find the surface area to the
nearest tenth if each side of the base is 12 centimeters and the height of the
pyramid is 8 centimeters.
Look at the pyramid above. The slant height is the hypotenuse of a right triangle. One leg of
that triangle is the height of the pyramid, and the other leg is half the length of a side of the
base. Use the Pythagorean Theorem to find the slant height .
2 62 82
100
10
Pythagorean Theorem
Simplify.
Take the square root of each side.
1
T P B
2
1
(4)(12)(10) 122
2
384
Surface area of a pyramid
P (4)(12), 10, B 122
Simplify.
The surface area is 384 square centimeters.
Exercises
Find the surface area of each regular pyramid. Round to the nearest tenth if
necessary.
1.
2.
8 ft
20 cm
45
15 cm
3.
4.
10 cm
8.7 in.
6 in.
60
5.
15 in.
6.
12 cm
13 cm
12 yd
10 yd
©
Glencoe/McGraw-Hill
686
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
12-5 Skills Practice
Surface Area of Pyramids
Find the surface area of each regular pyramid. Round to the nearest tenth if
necessary.
1.
2.
20 in.
7 cm
8 in.
4 cm
3.
4.
12 ft
9m
10 m
14 ft
5.
6.
9 mm
7 yd
6 yd
6 mm
7.
8.
12 m
18 m
©
Glencoe/McGraw-Hill
Lesson 12-5
20 in.
16 in.
687
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
12-6 Study Guide and Intervention
Lateral Areas of Cones
Cones have the following
V
V
axis
altitude
properties.
slant height
• A cone has one circular base and one vertex.
base
base
• The segment whose endpoints are the vertex and
oblique cone
right cone
the center of the base is the axis of the cone.
• The segment that has one endpoint at the vertex, is
perpendicular to the base, and has its other endpoint
on the base is the altitude of the cone.
• For a right cone the axis is also the altitude, and any segment from the circumference
of the base to the vertex is the slant height . If a cone is not a right cone, it is oblique.
Lateral Area
of a Cone
If a cone has a lateral area of L square units, a slant height of units,
and the radius of the base is r units, then L r .
Example
Find the lateral area of a cone with slant height of
10 centimeters and a base with a radius of 6 centimeters.
L r
Lateral area of a cone
(6)(10) r 6, 10
188.5
Simplify.
10 cm
6 cm
The lateral area is about 188.5 square centimeters.
Exercises
Find lateral area of each circular cone. Round to the nearest tenth.
1.
2.
15 cm
4 cm
3.
3 cm
9 cm
4.
60
26 mm
20 mm
6
3m
5.
12 in.
6.
8 yd
5 in.
©
Glencoe/McGraw-Hill
16 yd
691
Glencoe Geometry
Lesson 12-6
Surface Areas of Cones
NAME ______________________________________________ DATE
____________ PERIOD _____
12-6 Study Guide and Intervention
(continued)
Surface Areas of Cones
Surface Areas of Cones
The surface area of a cone is the
lateral area of the cone plus the area of the base.
Surface Area
of a Right Cone
If a cone has a surface area of T square units, a slant height of
units, and the radius of the base is r units, then T r r 2.
height
slant height
r
Example
For the cone above, find the surface area to the nearest tenth if the
radius is 6 centimeters and the height is 8 centimeters.
The slant height is the hypotenuse of a right triangle with legs of length 6 and 8. Use the
Pythagorean Theorem.
2 62 82 Pythagorean Theorem
2 100
Simplify.
10
Take the square root of each side.
2
T r r
Surface area of a cone
(6)(10) 62
301.6
r 6, 10
Simplify.
The surface area is about 301.6 square centimeters.
Exercises
Find the surface area of each cone. Round to the nearest tenth.
1.
12 cm
2.
5 ft
9 cm
30
3.
12 cm
4.
45
13 cm
4 in.
5.
26 m
6.
40 m
©
Glencoe/McGraw-Hill
8
3 yd
60
692
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
12-6 Skills Practice
Find the surface area of each cone. Round to the nearest tenth if necessary.
1.
2.
5m
10 ft
14 m
25 ft
3.
4.
21 in.
9 mm
17 mm
8 in.
5.
6.
7 cm
6 yd
22 cm
4 yd
7. Find the surface area of a cone if the height is 12 inches and the slant height is 15 inches.
8. Find the surface area of a cone if the height is 9 centimeters and the slant height is
12 centimeters.
9. Find the surface area of a cone if the height is 10 meters and the slant height is 14 meters.
10. Find the surface area of a cone if the height is 5 feet and the slant height is 7 feet.
©
Glencoe/McGraw-Hill
693
Glencoe Geometry
Lesson 12-6
Surface Areas of Cones
NAME ______________________________________________ DATE
____________ PERIOD _____
12-7 Study Guide and Intervention
Surface Areas of Spheres
Properties of Spheres
A sphere is the locus of all points that
are a given distance from a given point called its center.
center
V
chord
tangent
B
W
A
NR
diameter
S
T
X
M
great circle
radius
RS
is a radius. A
B
is a chord.
is a tangent.
T
S
is a diameter. VX
The circle that contains points S, M, T, and N is
a great circle; it determines two hemispheres.
Example
Determine the shapes you get when you intersect a
plane with a sphere.
O
M
P
The intersection of plane M
and sphere O is point P.
O
Q
N
The intersection of plane N
and sphere O is circle Q.
P
O
The intersection of plane P
and sphere O is circle O.
A plane can intersect a sphere in a point, in a circle, or in a great circle.
Exercises
Describe each object as a model of a circle, a sphere, a hemisphere, or none of these.
1. a baseball
2. a pancake
3. the Earth
4. a kettle grill cover
5. a basketball rim
6. cola can
Determine whether each statement is true or false.
7. All lines intersecting a sphere are tangent to the sphere.
8. Every plane that intersects a sphere makes a great circle.
9. The eastern hemisphere of Earth is congruent to the western hemisphere.
10. The diameter of a sphere is congruent to the diameter of a great circle.
©
Glencoe/McGraw-Hill
697
Glencoe Geometry
Lesson 12-7
Here are some terms associated with a sphere.
• A radius is a segment whose endpoints are the
center of the sphere and a point on the sphere.
• A chord is a segment whose endpoints are points
on the sphere.
• A diameter is a chord that contains the sphere’s
center.
• A tangent is a line that intersects the sphere in
exactly one point.
• A great circle is the intersection of a sphere and
a plane that contains the center of the sphere.
• A hemisphere is one-half of a sphere. Each great
circle of a sphere determines two hemispheres.
sphere
NAME ______________________________________________ DATE
____________ PERIOD _____
12-7 Study Guide and Intervention
(continued)
Surface Areas of Spheres
Surface Areas of Spheres
You can think of the surface area of a sphere
as the total area of all of the nonoverlapping strips it would take to cover the
sphere. If r is the radius of the sphere, then the area of a great circle of the
sphere is r2. The total surface area of the sphere is four times the area of a
great circle.
Surface Area
of a Sphere
r
If a sphere has a surface area of T square units and a radius of r units, then T 4r 2.
Example
Find the surface area of a sphere to the nearest tenth
if the radius of the sphere is 6 centimeters.
T 4r2
Surface area of a sphere
4 62 r 6
452.4
Simplify.
6 cm
The surface area is 452.4 square centimeters.
Exercises
Find the surface area of each sphere with the given radius or diameter to the
nearest tenth.
1. r 8 cm
2. r 22
ft
3. r cm
4. d 10 in.
5. d 6 m
6. d 16 yd
7. Find the surface area of a hemisphere with radius 12 centimeters.
8. Find the surface area of a hemisphere with diameter centimeters.
9. Find the radius of a sphere if the surface area of a hemisphere is
192 square centimeters.
©
Glencoe/McGraw-Hill
698
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
12-7 Skills Practice
Surface Areas of Spheres
In the figure, A is the center of the sphere, and plane T
intersects the sphere in circle E. Round to the nearest
tenth if necessary.
A
1. If AE 5 and DE 12, find AD.
D
E
2. If AE 7 and DE 15, find AD.
3. If the radius of the sphere is 18 units and the radius of E is 17 units, find AE.
4. If the radius of the sphere is 10 units and the radius of E is 9 units, find AE.
5. If M is a point on E and AD 23, find AM.
Find the surface area of each sphere or hemisphere. Round to the nearest tenth.
6.
7.
32 m
7 in.
8. a hemisphere with a radius of the great circle 8 yards
9. a hemisphere with a radius of the great circle 2.5 millimeters
10. a sphere with the area of a great circle 28.6 inches
©
Glencoe/McGraw-Hill
699
Glencoe Geometry
Lesson 12-7
T
NAME ______________________________________________ DATE
____________ PERIOD _____
12-7 Practice
Surface Areas of Spheres
In the figure, C is the center of the sphere, and plane B
intersects the sphere in circle R. Round to the nearest
tenth if necessary.
C
1. If CR 4 and SR 14, find CS.
S
R
B
2. If CR 7 and SR 24, find CS.
3. If the radius of the sphere is 28 units and the radius of R is 26 units, find CR.
4. If J is a point on R and CS 7.3, find CJ.
Find the surface area of each sphere or hemisphere. Round to the nearest tenth.
5.
6.
6.5 cm
89 ft
7. a sphere with the area of a great circle 29.8 meters
8. a hemisphere with a radius of the great circle 8.4 inches
9. a hemisphere with the circumference of a great circle 18 millimeters
10. SPORTS A standard size 5 soccer ball for ages 13 and older has a circumference of
27–28 inches. Suppose Breck is on a team that plays with a 28-inch soccer ball. Find the
surface area of the ball.
©
Glencoe/McGraw-Hill
700
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
13-1 Study Guide and Intervention
Volumes of Prisms and Cylinders
Volumes of Prisms The measure of the amount of space
that a three-dimensional figure encloses is the volume of the
figure. Volume is measured in units such as cubic feet, cubic
yards, or cubic meters. One cubic unit is the volume of a cube
that measures one unit on each edge.
cubic foot
cubic yard
27 cubic feet 1 cubic yard
If a prism has a volume of V cubic units, a height of h units,
and each base has an area of B square units, then V Bh.
Example 1
Find the volume
of the prism.
Example 2
Find the volume of the
prism if the area of each base is 6.3
square feet.
4 cm
base
3 cm
7 cm
V Bh
(7)(3)(4)
84
3.5 ft
Formula for volume
B (7)(3), h 4
Multiply.
The volume of the prism is 84 cubic
centimeters.
V Bh
(6.3)(3.5)
22.05
Formula for volume
B 6.3, h 3.5
Multiply.
The volume is 22.05 cubic feet.
Exercises
Find the volume of each prism. Round to the nearest tenth if necessary.
1.
2.
1.5 cm
8 ft
4 cm
8 ft
8 ft
3.
3 cm
4.
12 ft
12 ft
15 ft
5.
30
10 ft
15 ft
6.
2 cm
3 yd
1.5 cm
6 cm
7 yd
4 cm
©
Glencoe/McGraw-Hill
723
4 yd
Glencoe Geometry
Lesson 13-1
Volume
of a Prism
NAME ______________________________________________ DATE
____________ PERIOD _____
13-1 Study Guide and Intervention
(continued)
Volumes of Prisms and Cylinders
Volumes of Cylinders The volume of a cylinder is the product of the
height and the area of the base. The base of a cylinder is a circle, so the area
of the base is r2.
Volume of
a Cylinder
r
h
If a cylinder has a volume of V cubic units, a height of h units,
and the bases have radii of r units, then V r 2h.
Example 1
Find the volume
of the cylinder.
Example 2
Find the area of the
oblique cylinder.
3 cm
4 cm
13 in.
h
8 in.
5 in.
V r2h
(3)2(4)
113.1
Volume of a cylinder
r 3, h 4
Simplify.
The volume is about 113.1 cubic
centimeters.
The radius of each base is 4 inches, so the area of
the base is 16 in2. Use the Pythagorean Theorem
to find the height of the cylinder.
h2 52 132
h2 144
h 12
Pythagorean Theorem
Simplify.
Take the square root of each side.
V r2h
(4)2(12)
603.2 in3
Volume of a cylinder
r 4, h 12
Simplify.
Exercises
Find the volume of each cylinder. Round to the nearest tenth.
1.
2.
2 ft
2 cm
18 cm
1 ft
3.
4.
1.5 ft
12 ft
5.
20 ft
20 ft
6.
10 cm
1 yd
13 cm
©
Glencoe/McGraw-Hill
4 yd
724
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
13-1 Skills Practice
Volumes of Prisms and Cylinders
Find the volume of each prism or cylinder. Round to the nearest tenth if necessary.
1.
2.
2 ft
8 cm
8 ft
16 cm
6 ft
3.
4.
Lesson 13-1
18 cm
34 in.
13 m
5m
16 in.
22 in.
3m
5.
6.
23 mm
6 yd
10 yd
15 mm
Find the volume of each oblique prism or cylinder. Round to the nearest tenth if
necessary.
7.
4 cm
8.
18 cm
5 in.
17 cm
©
Glencoe/McGraw-Hill
3 in.
725
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
13-2 Study Guide and Intervention
Volumes of Pyramids and Cones
Volumes of Pyramids This figure shows a prism and a pyramid
that have the same base and the same height. It is clear that the volume
of the pyramid is less than the volume of the prism. More specifically,
the volume of the pyramid is one-third of the volume of the prism.
Volume of
a Pyramid
If a pyramid has a volume of V cubic units, a height of h units,
Example
1
V Bh
3
1
(8)(8)10
3
213.3
1
and a base with an area of B square units, then V 3Bh.
Find the volume of the square pyramid.
10 ft
Volume of a pyramid
B (8)(8), h 10
8 ft
8 ft
Multiply.
Exercises
Find the volume of each pyramid. Round to the nearest tenth if necessary.
1.
2.
10 ft
15 ft
6 ft
8 ft
12 ft
3.
10 ft
4.
12 cm
18 ft
8 cm
regular
hexagon
4 cm
5.
16 in.
6.
6 yd
8 yd
15 in.
5 yd
15 in.
©
Glencoe/McGraw-Hill
6 ft
729
Glencoe Geometry
Lesson 13-2
The volume is about 213.3 cubic feet.
NAME ______________________________________________ DATE
____________ PERIOD _____
13-2 Study Guide and Intervention
(continued)
Volumes of Pyramids and Cones
Volumes of Cones
For a cone, the volume is one-third the product of the
height and the base. The base of a cone is a circle, so the area of the base is r2.
Volume of a Right
Circular Cone
If a cone has a volume of V cubic units, a height of h units,
1
and the area of the base is B square units, then V 3Bh.
h
r
The same formula can be used to find the volume of oblique cones.
Example
Find the volume of the cone.
1
V r2h
3
1
(5)212
3
314.2
5 cm
Volume of a cone
12 cm
r 5, h 12
Simplify.
The volume of the cone is about 314.2 cubic centimeters.
Exercises
Find the volume of each cone. Round to the nearest tenth.
1.
2.
10 cm
6 cm
3.
10 ft
4.
18 yd 45
12 in.
30 in.
5.
20 yd
6.
26 ft
20 ft
©
8 ft
Glencoe/McGraw-Hill
45
16 cm
730
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
13-2 Skills Practice
Volumes of Pyramids and Cones
Find the volume of each pyramid or cone. Round to the nearest tenth if necessary.
1.
2.
8 cm
8 ft
5 ft
4 cm
7 cm
5 ft
3.
4.
12 m
14 in.
Lesson 13-2
25 m
8 in.
10 in.
5.
6.
14 yd
18 mm
66
25 yd
Find the volume of each oblique pyramid or cone. Round to the nearest tenth if
necessary.
7.
8.
6 cm
12 cm
6 ft
4 ft
4 ft
©
Glencoe/McGraw-Hill
731
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
13-3 Study Guide and Intervention
Volumes of Spheres
Volumes of Spheres A sphere has one basic measurement, the
length of its radius. If you know the radius of a sphere, you can calculate
its volume.
Volume of
a Sphere
r
If a sphere has a volume of V cubic units and a radius of r units, then V 4r 3.
3
Example 1
4
V r3
3
4
(8)3
3
2144.7
Find the volume of a sphere with radius 8 centimeters.
8 cm
Volume of a sphere
r8
Simplify.
The volume is about 2144.7 cubic centimeters.
Example 2
A sphere with radius 5 inches just fits inside
a cylinder. What is the difference between the volume of the
cylinder and the volume of the sphere? Round to the nearest
cubic inch.
The base of the cylinder is 25 in2 and the height is 10 in., so the
4
volume of the cylinder is 250 in3. The volume of the sphere is (5)3
500
3
500
3
5 in.
5 in.
5 in.
5 in.
3
Lesson 13-3
or in3. The difference in the volumes is 250 or about 262 in3.
Exercises
Find the volume of each solid. Round to the nearest tenth.
1.
2.
3.
6 in.
5 ft
4.
16 in.
5.
5 in.
13 in.
8 cm
6.
8 in.
difference
between
volume of cube
and volume
of sphere
7. A hemisphere with radius 16 centimeters just fits inside a rectangular prism. What is
the difference between the volume of the prism and the volume of the hemisphere?
Round to the nearest cubic centimeter.
©
Glencoe/McGraw-Hill
735
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
13-3 Study Guide and Intervention
(continued)
Volumes of Spheres
Solve Problems Involving Volumes of Spheres If you want to know if a sphere
can be packed inside another container, or if you want to compare the capacity of a sphere
and another shape, you can compare volumes.
Example
Compare the volumes of the sphere and
the cylinder. Determine which quantity is greater.
4
3
V r3
Volume of sphere
V r2h
r2(1.5r)
1.5r3
r
Volume of cylinder
1.5r
h 1.5r
Simplify.
4
4
Compare r3 with 1.5r3. Since is less than 1.5, it follows that
3
3
the volume of the sphere is less than the volume of the cylinder.
Exercises
Compare the volume of a sphere with radius r to the volume of each figure below.
Which figure has a greater volume?
1.
2.
2r
r
3.
r
2
4.
r
r
r
r
r
5.
3r
6.
3r
2a
r
©
Glencoe/McGraw-Hill
736
Glencoe Geometry
NAME ______________________________________________ DATE
____________ PERIOD _____
13-3 Skills Practice
Volumes of Spheres
Find the volume of each sphere or hemisphere. Round to the nearest tenth.
1. The radius of the sphere is 9 centimeters.
2. The diameter of the sphere is 10 inches.
3. The circumference of the sphere is 26 meters.
4. The radius of the hemisphere is 7 feet.
5. The diameter of the hemisphere is 12 kilometers.
7.
Lesson 13-3
6. The circumference of the hemisphere is 48 yards.
8.
94.8 ft
16.2 cm
9.
10.
4.5 in.
14.4 m
©
Glencoe/McGraw-Hill
737
Glencoe Geometry