3.4c:Surface Area and Volume of Spheres

M(G&M)–10–6 Solves problems involving perimeter, circumference, or area of two dimensional figures (including composite figures) or surface area or volume of three

GSE’s

G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.?

CCSS:

Finding the Surface Area of a Sphere

• Sphere- all of the points in space that are the same distance from a point.

How is this different from a circle?

Finding the Surface Area of a Sphere

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

Surface Area of a Sphere

• The surface area of a sphere with radius r is S = 4r2.

Finding the Surface Area of a Sphere

• 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.

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.

Ex. 2: Using a Great Circle

• The circumference of a great circle of a sphere 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 = 2r

13.8 2r

6.9 = r

= 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 ft.2

Theorem Volume of a Sphere

• The volume of a sphere with radius r is V =

33

3

4r

Find the Volume of a sphere with a radius of 5 in.

Find the total surface area and Volume

The circumference of the cone is 3 inches

The cone is completely packed with Ice cream, and there is exactly half a scoop on top.

Find how much ice cream you will receive.