12.6 Surface Area & Volume 12.6 Surface Area & Volume of Spheresof Spheres

DefinitionsDefinitions• SphereSphere – the locus of points in – the locus of points in spacespace that that

are a given distance from a given point. are a given distance from a given point. (looks like a ball)(looks like a ball)

• Center of a SphereCenter of a Sphere – the given point in the – the given point in the middle.middle.

• Radius of a SphereRadius of a Sphere – segment from the – segment from the center to a point on the sphere.center to a point on the sphere.

• Chord of a SphereChord of a Sphere – a segment whose – a segment whose endpoints are on the sphere.endpoints are on the sphere.

• Diameter of a SphereDiameter of a Sphere – a chord that goes – a chord that goes through the center.through the center.

Parts of a SphereParts of a SphereC is the center of the C is the center of the

sphere.sphere.

AB is a diameter.AB is a diameter.

CB & AC are radii.CB & AC are radii.

DE & AB are chords.DE & AB are chords.CC

AA BB

DDEE

Thm 12.11Thm 12.11 – Surface Area of a Sphere – Surface Area of a Sphere

S = 4S = 4rr22

(it takes 4 circles to cover a sphere)(it takes 4 circles to cover a sphere)

Why isn’t there a lateral area formula?Why isn’t there a lateral area formula?

Because spheres have no bases!Because spheres have no bases!

ExEx: Find the surface area of a sphere : Find the surface area of a sphere with a diameter of 8 cm.with a diameter of 8 cm.

S = 4S = 4(4)(4)22

S = 4S = 4(16)(16)

S = 64S = 64 cm cm22

More DefinitionsMore Definitions

• Great Circle of a SphereGreat Circle of a Sphere – the cross section – the cross section of a sphere sliced by a plane through its of a sphere sliced by a plane through its center.center.

• HemisphereHemisphere – ½ of a sphere. – ½ of a sphere.

** Every great circle splits a sphere into 2 ** Every great circle splits a sphere into 2 hemispheres.hemispheres.

ExEx: The circumference of a great : The circumference of a great circle of a sphere is 16circle of a sphere is 16 m. What is m. What is

the surface area of the sphere?the surface area of the sphere?

C = 2C = 2rr

1616 = 2 = 2rr

16 = 2r16 = 2r

8 m = r8 m = r

S = 4S = 4rr22

S = 4S = 4(8)(8)22

S = 4S = 4(64)(64)

S = 256S = 256 m m22

Thm 12.12Thm 12.12 – Volume of a Sphere – Volume of a Sphere

ExEx: Find the volume of a sphere with a : Find the volume of a sphere with a radius of 3 ft.radius of 3 ft.

V = 36V = 36 ft ft33

3

3

4rV

333

4V 273

4