Glencoe Geometry - Burlington County Institute of Technology

Report
Five-Minute Check (over Lesson 12–5)
CCSS
Then/Now
New Vocabulary
Key Concept: Surface Area of a Sphere
Example 1: Surface Area of a Sphere
Example 2: Use Great Circles to Find Surface Area
Key Concept: Volume of a Sphere
Example 3: Volumes of Spheres and Hemispheres
Example 4: Real-World Example: Solve Problems Involving
Solids
Over Lesson 12–5
Find the volume of the cone. Round to the nearest
tenth if necessary.
A. 134.0 mm3
B. 157.0 mm3
C. 201.1 mm3
D. 402.1 mm3
Over Lesson 12–5
Find the volume of the pyramid. Round to the
nearest tenth if necessary.
A. 36 ft3
B. 125 ft3
C. 180 ft3
D. 270 ft3
Over Lesson 12–5
Find the volume of the cone. Round to the nearest
tenth if necessary.
A. 323.6 ft3
B. 358.1 ft3
C. 382.5 ft3
D. 428.1 ft3
Over Lesson 12–5
Find the volume of the pyramid. Round to the
nearest tenth if necessary.
A. 1314.3 in3
B. 1177.0 in3
C. 1009.4 in3
D. 987.5 in3
Over Lesson 12–5
Find the volume of a cone with a diameter of
8.4 meters and a height of 14.6 meters.
A. 192.6 m3
B. 237.5 m3
C. 269.7 m3
D. 385.2 m3
Over Lesson 12–5
Find the height of a hexagonal pyramid with a
base area of 130 square meters and a volume of
650 cubic meters.
A. 12 m
B. 15 m
C. 17 m
D. 22 m
Content Standards
G.GMD.1 Give an informal argument for the
formulas for the circumference of a circle, area of a
circle, volume of a cylinder, pyramid, and cone.
G.GMD.3 Use volume formulas for cylinders,
pyramids, cones, and spheres to solve problems.
Mathematical Practices
1 Make sense of problems and persevere in solving
them.
6 Attend to precision.
You found surface areas of prisms and
cylinders.
• Find surface areas of spheres.
• Find volumes of spheres.
• great circle
• pole
• hemisphere
Surface Area of a Sphere
Find the surface area of the
sphere. Round to the nearest
tenth.
S = 4r2
Surface area of a sphere
= 4(4.5)2
Replace r with 4.5.
≈ 254.5
Simplify.
Answer: 254.5 in2
Find the surface area of the sphere. Round to the
nearest tenth.
A. 462.7 in2
B. 473.1 in2
C. 482.6 in2
D. 490.9 in2
Use Great Circles to Find Surface Area
A. Find the surface area of the hemisphere.
Find half the area of a sphere with the radius of
3.7 millimeters. Then add the area of the great circle.
Use Great Circles to Find Surface Area
Surface area of a
hemisphere
Replace r with 3.7.
≈ 129.0
Answer: about 129.0 mm2
Use a calculator.
Use Great Circles to Find Surface Area
B. Find the surface area of a sphere if the
circumference of the great circle is 10 feet.
First, find the radius. The circumference of a great
circle is 2r. So, 2r = 10 or r = 5.
Use Great Circles to Find Surface Area
S = 4r2
Surface area of a sphere
= 4(5)2
Replace r with 5.
≈ 314.2
Use a calculator.
Answer: about 314.2 ft2
Use Great Circles to Find Surface Area
C. Find the surface area of a sphere if the area of
the great circle is approximately 220 square
meters.
First, find the radius. The area of a great circle is r2.
So, r2 = 220 or r ≈ 8.4.
Use Great Circles to Find Surface Area
S = 4r2
Surface area of a sphere
≈ 4(8.4)2
Replace r with 5.
≈ 886.7
Use a calculator.
Answer: about 886.7 m2
A. Find the surface area of the hemisphere.
A. 110.8 m2
B. 166.3 m2
C. 169.5 m2
D. 172.8 m2
B. Find the surface area of a sphere if the
circumference of the great circle is 8 feet.
A. 100.5 ft2
B. 201.1 ft2
C. 402.2 ft2
D. 804.3 ft2
C. Find the surface area of the sphere if the area of
the great circle is approximately 160 square
meters.
A. 320 ft2
B. 440 ft2
C. 640 ft2
D. 720 ft2
Volumes of Spheres and Hemispheres
A. Find the volume a sphere with a great circle
circumference of 30 centimeters. Round to the
nearest tenth.
Find the radius of the sphere. The
circumference of a great circle is
2r. So, 2r = 30 or r = 15.
Volume of a sphere
(15)3
r = 15
≈ 14,137.2 cm3 Use a calculator.
Volumes of Spheres and Hemispheres
Answer: The volume of the sphere is approximately
14,137.2 cm3.
Volumes of Spheres and Hemispheres
B. Find the volume of the
hemisphere with a diameter of
6 feet. Round to the nearest tenth.
The volume of a hemisphere is
one-half the volume of the sphere.
Volume of a hemisphere
r
3
Use a calculator.
Answer: The volume of the hemisphere is
approximately 56.5 cubic feet.
A. Find the volume of the sphere to the nearest
tenth.
A. 268.1 cm3
B. 1608.5 cm3
C. 2144.7 cm3
D. 6434 cm3
B. Find the volume of the hemisphere to the nearest
tenth.
A. 3351.0 m3
B. 6702.1 m3
C. 268,082.6 m3
D. 134,041.3 m3
Solve Problems Involving Solids
ARCHEOLOGY The stone spheres of Costa Rica
were made by forming granodiorite boulders into
spheres. One of the stone spheres has a volume of
about 36,000 cubic inches. What is the diameter
of the stone sphere?
Understand You know that the volume of the stone
is 36,000 cubic inches.
Plan
First use the volume formula to find the
radius. Then find the diameter.
Solve Problems Involving Solids
Solve
Volume of a sphere
Replace V with 36,000.
Divide each side by
Use a calculator to find
2700
(
1
÷
3
)
ENTER
30
The radius of the stone is 30 inches. So, the diameter
is 2(30) or 60 inches.
Solve Problems Involving Solids
Answer: 60 inches
CHECK You can work backward to check the
solution.
If the diameter is 60, then r = 30. If r = 30,
then V =
The solution is correct. 
cubic inches.
RECESS The jungle gym outside of Jada’s school
is a perfect hemisphere. It has a volume of 4,000
cubic feet. What is the diameter of the jungle gym?
A. 10.7 feet
B. 12.6 feet
C. 14.4 feet
D. 36.3 feet

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