12.6 SurfaceArea and Volumeof SpheresGeometryCreated By Mrs. Spitz
Objectives/Assignment• Find the surface area of a sphere.• Find the volume of a sphere in reallife such as the ball bearing.
Finding the Surface Area of aSphere• In Lesson 10.7, a circle was described asa locus of points in a plane that are agiven distance from a point. A sphere isthe locus of points in space that are agiven distance from a point.
Finding the Surface Area of a Sphere• The point is called the center of thesphere. A radius of a sphere is asegment from the center to a pointon the sphere.• A chord of a sphere is a segmentwhose endpoints are on the sphere.
Finding the Surface Area of a Sphere• A diameter is a chord that containsthe center. As with all circles, theterms radius and diameter alsorepresent distances, and thediameter is twice the radius.
Theorem Surface Area of a Sphere• The surface area of a sphere withradius r is S = 4πr2.
Ex. 1: Finding the SurfaceArea of a Sphere• Find the surface area. When theradius doubles, does the surfacearea double?
S = 4πr2= 4π22= 16π in.2S = 4πr2= 4π42= 64π in.2The surface area of the sphere in part (b) is fourtimes greater than the surface area of the sphere inpart (a) because 16π • 4 = 64πSo, when the radius of a sphere doubles, thesurface area DOES NOT double.
More . . .• If a plane intersects a sphere, theintersection is either a single point ora circle. If the plane contains thecenter of the sphere, then theintersection is a great circle of thesphere. Every great circle of asphere separates a sphere into twocongruent halves calledhemispheres.
Ex. 2: Using a Great Circle• The circumference of a great circleof a sphere is 13.8π feet. What isthe surface area of the sphere?
Solution:Begin by finding the radius of thesphere.C = 2πr13.8π = 2πr13.8π2πr6.9 = r= r
Solution:Using a radius of 6.9 feet, the surfacearea is:S = 4πr2= 4π(6.9)2= 190.44π ft.2So, the surface area of the sphere is190.44 π ft.2
Ex. 3: Finding the SurfaceArea of a Sphere• Baseball. A baseball and its leathercovering are shown. The baseball has aradius of about 1.45 inches.a. Estimate the amount of leather used tocover the baseball.b. The surface area of a baseball is sewnfrom two congruent shapes, each whichresembles two joined circles. How doesthis relate to the formula for the surfacearea of a sphere?
Theorem Volume of a Sphere• The volume of a sphere with radius ris S = 4πr3.3
Ex. 4: Finding the Volume of aSphere• Ball Bearings. To make asteel ball bearing, acylindrical slug is heatedand pressed into aspherical shape with thesame volume. Find theradius of the ball bearingto the right:
Solution:• To find the volume of the slug, use theformula for the volume of a cylinder.V = πr2h= π(12)(2)= 2π cm3To find the radius of the ball bearing, use theformula for the volume of a sphere andsolve for r.