Geometric Solids

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Spheres

Spheres

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A sphere is formed by revolving a circle about its diameter.

In space, the set of all points that are a given distance from a given point, called the center.

Definition:

Spheres and Circle Terminology

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Radius: A segment whose endpoints are the center of the sphere and a point on the sphere.

Chord: A segment whose endpoints are on the sphere.Diameter: A chord that contains the sphere’s center.Tangent: A line that intersects the sphere in exactly one point.

Radius

Chord

DiameterTangen

t

Surface Area & Volume of Sphere

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3r

2 2

3 3

. 4 12 576

412 2304

3

S A cm

V cm

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Volume (V) =

Surface Area (SA) = 4 π r2

r

Example:

Find the surface area and volume of the sphere.

12 cm

Great Circle & Hemisphere

Great Circle: For a given sphere, the intersection of the sphere and a plane that contains the center of the sphere. A geographic representation would be the equator.Hemisphere: One of the two parts into which a great circle separates a given sphere.

Great Circle

Hemisphere

Finding the Volume of a Hemisphere

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3

4. 10 1333.3

31333.3

. 666.72

V of Sphere

V of Hemisphere cm

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Find the volume of the following solid (Hemisphere).

10 cm

Finding the Surface Area of a Hemisphere

2 2

2

2 2

. . 4 10 400

400. . 200

2

200

200 (10) 200 100 300

S A of Sphere cm

S A of Hemisphere

Area of Great Circle r

cm

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Find the Surface Area of the following solid (Hemisphere).

10 cm

Example #1

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The beach ball has a radius of 7 inches. Find the Surface Area of the ball.

What if we wanted to fill the ball up with water? How would we determine how much water the ball could hold? Find the

volume!.V

4

3r3

4

3 (73 )

4

3 (343)1372

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Example #2

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Example #2 Solution

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Example #3

A solid metal right cylinder is melted and recast as a sphere. The radius of the cylinder was 4 and the height was 18. Find the volume and surface area of the sphere.

The volume will not change! Start there.

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Example #3 continued

Since the Volume of the Cylinder is the same as the Volume of the Sphere, then

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