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

A. 134.0 mm3

B. 157.0 mm3

C. 201.1 mm3

D. 402.1 mm3

Find the volume of the cone. Round to the nearest tenth if necessary.

Over Lesson 12–5

A. 36 ft3

B. 125 ft3

C. 180 ft3

D. 270 ft3

Find the volume of the pyramid. Round to the nearest tenth if necessary.

Over Lesson 12–5

A. 323.6 ft3

B. 358.1 ft3

C. 382.5 ft3

D. 428.1 ft3

Find the volume of the cone. Round to the nearest tenth if necessary.

Over Lesson 12–5

A. 1314.3 in3

B. 1177.0 in3

C. 1009.4 in3

D. 987.5 in3

Find the volume of the pyramid. Round to the nearest tenth if necessary.

Over Lesson 12–5

A. 192.6 m3

B. 237.5 m3

C. 269.7 m3

D. 385.2 m3

Find the volume of a cone with a diameter of 8.4 meters and a height of 14.6 meters.

Over Lesson 12–5

A. 12 m

B. 15 m

C. 17 m

D. 22 m

Find the height of a hexagonal pyramid with a base area of 130 square meters and a volume of 650 cubic meters.

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 = 4r2 Surface area of a sphere

= 4(4.5)2 Replace r with 4.5.

≈ 254.5 Simplify.

Answer: 254.5 in2

A. 462.7 in2

B. 473.1 in2

C. 482.6 in2

D. 490.9 in2

Find the surface area of the sphere. Round to the nearest tenth.

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

Answer: about 129.0 mm2

Replace r with 3.7.

Use a calculator.≈ 129.0

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 2r. So, 2r = 10 or r = 5.

Use Great Circles to Find Surface Area

Answer: about 314.2 ft2

S = 4r2 Surface area of a sphere

= 4(5)2 Replace r with 5.

≈ 314.2 Use a calculator.

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

Answer: about 886.7 m2

S = 4r2 Surface area of a sphere

≈ 4(8.4)2 Replace r with 5.

≈ 886.7 Use a calculator.

A. 110.8 m2

B. 166.3 m2

C. 169.5 m2

D. 172.8 m2

A. Find the surface area of the hemisphere.

A. 100.5 ft2

B. 201.1 ft2

C. 402.2 ft2

D. 804.3 ft2

B. Find the surface area of a sphere if the circumference of the great circle is 8 feet.

A. 320 ft2

B. 440 ft2

C. 640 ft2

D. 720 ft2

C. Find the surface area of the sphere if the area of the great circle is approximately 160 square meters.

Volumes of Spheres and Hemispheres

A. Find the volume a sphere with a great circle circumference of 30 centimeters. Round to the nearest tenth.

Volume of a sphere

(15)3 r = 15

≈ 14,137.2 cm3 Use a calculator.

Find the radius of the sphere. The circumference of a great circle is 2r. So, 2r = 30 or r = 15.

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.

Answer: The volume of the hemisphere is approximately 56.5 cubic feet.

Volume of a hemisphere

Use a calculator.

r 3

A. 268.1 cm3

B. 1608.5 cm3

C. 2144.7 cm3

D. 6434 cm3

A. Find the volume of the sphere to the nearest tenth.

A. 3351.0 m3

B. 6702.1 m3

C. 268,082.6 m3

D. 134,041.3 m3

B. Find the volume of the hemisphere to the nearest tenth.

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 stoneis 36,000 cubic inches.

Plan First use the volume formula to find theradius. Then find the diameter.

Solve Problems Involving Solids

Replace V with 36,000.

Solve Volume of a sphere

Divide each side by

Use a calculator to find

÷ ENTER( )2700 1 3 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 thesolution.

If the diameter is 60, then r = 30. If r = 30,

then V = cubic

inches.

The solution is correct.

A. 10.7 feet

B. 12.6 feet

C. 14.4 feet

D. 36.3 feet

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?