Surface Areas and Volumes of Spheres LESSON 12–6

Lesson Menu Five-Minute Check (over Lesson 12–5) TEKS 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 5-Minute Check 1 A mm 3 B mm 3 C mm 3 D mm 3 Find the volume of the cone. Round to the nearest tenth if necessary.

Over Lesson 12–5 5-Minute Check 2 A.36 ft 3 B.125 ft 3 C.180 ft 3 D.270 ft 3 Find the volume of the pyramid. Round to the nearest tenth if necessary.

Over Lesson 12–5 5-Minute Check 3 A ft 3 B ft 3 C ft 3 D ft 3 Find the volume of the cone. Round to the nearest tenth if necessary.

Over Lesson 12–5 5-Minute Check 4 A in 3 B in 3 C in 3 D in 3 Find the volume of the pyramid. Round to the nearest tenth if necessary.

Over Lesson 12–5 5-Minute Check 5 A m 3 B m 3 C m 3 D m 3 Find the volume of a cone with a diameter of 8.4 meters and a height of 14.6 meters.

Over Lesson 12–5 5-Minute Check 6 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.

TEKS Targeted TEKS G.11(C) Apply the formulas for the total and lateral surface area of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure. G.11(D) Apply the formulas for the volume of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems using appropriate units of measure. Also addresses G.10(B). Mathematical Processes G.1(A), G.1(E)

Then/Now You found surface areas of prisms and cylinders. Find surface areas of spheres. Find volumes of spheres.

Vocabulary great circle pole hemisphere

Concept

Example 1 Surface Area of a Sphere Find the surface area of the sphere. Round to the nearest tenth. S=4  r 2 Surface area of a sphere =4  (4.5) 2 Replace r with 4.5. ≈254.5Simplify. Answer:254.5 in 2

Example 1 A in 2 B in 2 C in 2 D in 2 Find the surface area of the sphere. Round to the nearest tenth.

Example 2A 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.

Example 2A Use Great Circles to Find Surface Area Surface area of a hemisphere Answer: about mm 2 Replace r with 3.7. Use a calculator. ≈ 129.0

Example 2B 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.

Example 2B Use Great Circles to Find Surface Area Answer: about ft 2 S=4  r 2 Surface area of a sphere =4  (5) 2 Replace r with 5. ≈314.2Use a calculator.

Example 2C 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  r 2. So,  r 2 = 220 or r ≈ 8.4.

Example 2C Use Great Circles to Find Surface Area Answer: about m 2 S=4  r 2 Surface area of a sphere ≈4  (8.4) 2 Replace r with 5. ≈886.7Use a calculator.

Example 2A A m 2 B m 2 C m 2 D m 2 A. Find the surface area of the hemisphere.

Example 2B A ft 2 B ft 2 C ft 2 D ft 2 B. Find the surface area of a sphere if the circumference of the great circle is 8  feet.

Example 2C A.320 ft 2 B.440 ft 2 C.640 ft 2 D.720 ft 2 C. Find the surface area of the sphere if the area of the great circle is approximately 160 square meters.

Concept

Example 3A 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 cm 3 Use a calculator. Find the radius of the sphere. The circumference of a great circle is 2  r. So, 2  r = 30  or r = 15.

Example 3A Volumes of Spheres and Hemispheres Answer: The volume of the sphere is approximately 14,137.2 cm 3.

Example 3B 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

Example 3A A cm 3 B cm 3 C cm 3 D.6434 cm 3 A. Find the volume of the sphere to the nearest tenth.

Example 3B A m 3 B m 3 C.268,082.6 m 3 D.134,041.3 m 3 B. Find the volume of the hemisphere to the nearest tenth.

Example 4 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? AnalyzeYou know that the volume of the stone is 36,000  cubic inches. FormulateFirst use the volume formula to find the radius. Then find the diameter.

Example 4 Solve Problems Involving Solids Replace V with 36,000 . DetermineVolume of a sphere Divide each side by Use a calculator to find ÷ENTER() The radius of the stone is 30 inches. So, the diameter is 2(30) or 60 inches.

Example 4 Solve Problems Involving Solids Answer: 60 inches JustifyYou can work backward to check the solution. If the diameter is 60, then r = 30. If r = 30, then V = cubic inches. The solution is correct. EvaluateSince the stone was a sphere allows use of the formula for the volume of a sphere to find a solution that closely approximates the actual diameter.

Example 4 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?

Surface Areas and Volumes of Spheres LESSON 12–6