Surface Area and Volume of Spheres

• A sphere is the set of all points in space that are the same distance from a point, the center of the sphere.

• A radius of a sphere is a segment from the center to a point on the sphere.

• A chord of a sphere is a segment whose endpoints are on the sphere.

Surface Area and Volume of Spheres

• A diameter is a chord that contains the center. As with all circles, the terms radius and diameter also represent distances, and the diameter is twice the radius.

Surface Area and Volume of Spheres

More . . .

• If a plane intersects a sphere, the intersection is either a single point or a circle. If the plane contains the center of the sphere, then the intersection is a great circle of the sphere. Every great circle of a sphere separates a sphere into two congruent halves called hemispheres.

Surface Area of a Sphere

• Surface area = 4 π (radius)2

• S = 4 π r2

C Radius

Find the Surface Area of a Sphere• Find the surface area of the sphere. Round your

answer to the nearest whole number.

The radius is 8 in.

24S r24 (8)

24 64 804 .in

8 in.

Find the Surface Area of a Sphere

• Find the surface area of the sphere. Round your answer to the nearest whole number.

The radius is 5 cm.

24S r24 (5)

24 25 314 .cm

10 cm.

Ex. 1: Comparing Surface Areas

• Find the surface area. When the radius doubles, does the surface area double?

S = 4r2

= 422

= 16 in.2

S = 4r2

= 442

= 64 in.2

The surface area of the sphere in part (b) is four times greater than the surface area of the sphere in part (a) because 16 • 4 = 64

So, when the radius of a sphere doubles, the surface area DOES NOT double.

Ex. 2: Using a Great Circle

• The circumference of a sphere (also referred to as the circumference of the great circle) is 13.8 feet. What is the surface area of the sphere?

Solution:

Begin by finding the radius of the sphere.

C = 2r

13.8 = 2r13.8 = r (To solve for r, divide both sides by 2)

2 6.9 = r

Solution:

Using a radius of 6.9 feet, the surface area is:

S = 4r2

= 4(6.9)2

= 190.44 ft.2

So, the surface area of the sphere is 190.44 feet squared.

Ex. 3: Finding the Surface Area of a Sphere

Volume of a Sphere

34

3V r

radius

C

Find the Volume of a Sphere• Find the volume of the sphere. Round your answer to

the nearest whole number.

a. 34

3V r

34(2)

3

334 .ft

2 ft.

4 Ab/c 3 2 ^ 3 =

Using your calculator, input:

Find the Volume of a Sphere

Find the volume of the hemisphere.

Round your answer to the nearest whole number.

Note: A hemisphere has HALF

the volume of a sphere.

• So simply use the volume formula,

then ½ the answer.

3262 .in

a5 in.

Using your calculator, input: 4 Ab/c 3 5 ^ 3 =

34

3V r

≈523.95877 / 2 = ≈262

Find the Volume of a Sphere

• Find the volume of the sphere. Write your answer in terms of .

34

3V r

34(6)

3

2883905 .in

6 in.

Be sure to remember to include when you write your answer

288x

Last Example: Find r, if V = 2V = 4 r3

3

2 = 4 r3

3

6 = 4r3

1.5 = r3

1.14 r

Formula for volume of a sphere.

Substitute 2 for V.

Multiply each side by 3.

Divide each side by 4.

Use a calculator to take the cube root.

The End

• Assignment: 6.9 NTG p. 245 – 251 all

• VOCABULARY on p. 245 –Sketch picture or write definition.

• NOTES PAGES p. 246-248 – Read and fill in the blanks.

• PRACTICE p. 249-251 – Complete 1 – 33 all.

• If you get stuck, refer back to your Presentation Notes.