find each measurement. 1. the radius of circle m if the diameter is 25 cm

MM2G4 Students will use apply surface area and volume of a sphere.

MM2G4 a Use and apply the area and volume of a sphere.MM2G4 b Determine the effect on surface area and volume of changing the radius or diameter of a sphere.

Find each measurement.

1. the radius of circle M if the diameter is 25 cm

2. the circumference of circle X if the radius is

42.5 in.

3. the area of circle T if the diameter is 26 ft

4. the circumference of circle N if the area is

625 cm2

12.5 cm

85 in.

169 ft2

50 cm

Warm Ups

MM2G4 Students will use apply surface area and volume of a sphere.


Surface Area and Volume of Spheres

How do we find the surface area and volume of a sphere?

Thursday, April 20, 2023

M2 Unit 3: Day 12

Lesson 6.9

MM2G4 Students will find and compare the measures of a sphere.


sphere

center of a sphere

radius of a sphere

hemisphere

great circle

Vocabulary



A sphere is the set of points in space that are a fixed distance from a given point called the center of a sphere. A radius of a sphere connects the center of the sphere to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres



Vocabulary

Great Circle – The intersection of a sphere and a plane that contains the center of the sphere.

Hemisphere – half of a sphere, formed when a great circle separates a sphere into two congruent halves.



EXAMPLE Find the surface area of a sphere

SOLUTION

S = 4πr2

= 4π(82)

= 256π

≈ 804.25

Formula for surface area of a sphere.

Substitute 8 for r.

Simplify.

Use a calculator.

Find the surface area of the sphere.

The surface area of the sphere is about 804.25 squareinches.

ANSWER



EXAMPLE 2 Standardized Test Practice

SOLUTION S = 4πr2

20.25π = 4πr2

5.0625 = r2

2.25 = r


Substitute 20.25π for S.

Divide each side by 4π.

Find the positive square root.

The diameter of the sphere is 2r = 2 2.25 = 4.5

centimeters.The correct answer is B. ANSWER



Finding Surface Area of Spheres

1. Find the surface area of a sphere with diameter 76 cm. Give your answers in terms of .

S = 4r2

S = 4(38)2 = 5776 cm2

Surface area of a sphere

Guided Practice



Finding Surface Area of Spheres

2. Find the surface area of a sphere with a great circle that has an area of 49 mi2.

Substitute 49 for A.49 = r2

r = 7 Solve for r.

S = 4r2

= 4(7)2 = 196 mi2 Substitute 7 for r.

A = r2 Area of a circle

Guided Practice



3. Find the surface area of the sphere.

Substitute 25 for r.

S = 2500 cm2

S = 4r2

S = 4(25)2

Surface area of a sphere

Guided Practice



GUIDED PRACTICE

4. The diameter of a sphere is 40 feet. Find the surface area of the sphere.

S = 4πr2

= 4πr(20)2

= 5026.55



Use a calculator.

The surface area of the sphere is about 5026.55 feet2

ANSWER

Guided Practice

= 1600π



GUIDED PRACTICE

5. The surface area of a sphere is 30π square meters.Find the radius of the sphere.

S = 4πr2

30π = 4πr2

7.5 = r2

2.74 = r


Substitute 30π for S.

Divide each side by 4π.

Find the positive square root.

Guided Practice



EXAMPLE 3

Use the circumference of a sphere

6. EXTREME SPORTS

In a sport called sphereing, a person rolls down a hill inside an inflatable ball surrounded by another ball. The diameter of the outer ball is 12 feet. Find the surface area of the outer ball.

SOLUTION

The diameter of the outer sphere is 12 feet, so the

radius is = 6 feet.12

2Use the formula for the surface area of a sphere.

S = 4πr2 = 4π(62) = 144πThe surface area of the outer ball is 144π, or about 452.39 square feet.ANSWER

Guided Practice



EXAMPLE 4Find the volume of a sphere

SOLUTION

The soccer ball has a diameter of 9 inches. Find its volume.

Formula for volume of a sphere

Substitute.

V = πr34

3

= π(4.5)343

The diameter of the ball is 9 inches, so the radius is

= 4.5 inches.9

2

= 121.5π

≈ 381.70

Simplify.

Use a calculator.

The volume of the soccer ball is 121.5π, or about 381.70 cubic inches.

ANSWER

EXAMPLE



Finding Volumes of Spheres

7. Find the volume of the sphere. Give your answer in terms of .

= 2304 in3 Simplify.

Volume of a sphere.

Guided Practice




8. Find the diameter of a sphere with volume 36,000 cm3.

Substitute 36,000 for V.

27,000 = r3

r = 30

d = 60 cm d = 2r

Take the cube root of both sides.

Volume of a sphere.

Guided Practice



9. Find the radius of a sphere with volume 2304 ft3.

Volume of a sphere

Substitute for V.

r = 12 ft Simplify.

Guided Practice



10. Find the volume of a sphere with surface area 324 in2. Give your answers in terms of .

Substitute 324 for S.324 = 4r2

r = 9 Solve for r.

Substitute 9 for r.

The volume of the sphere is 972 in3.

S = 4r2 Surface area of a sphere

Guided Practice




11. Find the volume of the hemisphere.

Volume of a hemisphere


= 2250 m3 Simplify.

Guided Practice



The radius of the sphere is divided by 3. Describe the effect on the surface area.

original dimensions: dimensions divided by 3:

The surface area is divided by 9.

S = 4r2

= 4(3)2 = 36 m3

S = 4r2

= 4(1)2 = 4 m3

Exploring Effects of Changing Dimensions

EXAMPLE



Sports Application

A sporting goods store sells exercise balls in two sizes, standard (12-in. diameter) and jumbo (36-in. diameter). How many times as great is the volume of a jumbo ball as the volume of a standard ball?

standard ball: jumbo ball:

A jumbo ball is about (3)3 = 27 times as great in volume as a standard ball.

Guided Practice

12. Exploring Effects of Changing Dimensions

34

3V rp=

( )346

3V p=

3288V inp=

34

3V rp=

( )3418

3V p=

37776V inp=



Homework 239 # 2 – 16 even.17,18

find each measurement. 1. the radius of circle m if the diameter is 25 cm

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