12.6 Surface Area and Volume of Spheres Geometry 12.6 Surface Area and Volume of Spheres mbhaub@mpsaz.org

Geometry 11.7 Surface Area and Volume of Spheres 11.7 Essential Question How do you find the surface area and volume of a sphere? November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Goals Find the surface area of spheres. Find the volume of spheres. Solve problems using area and volume. November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres The set of points in space that are equidistant from the same point, the center. radius Great Circle (Divides the sphere into two halves.) November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Hemisphere Half of a sphere. r November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Sphere Formulas r November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Using a Calculator You may find it easier to use the formula for volume in this form: November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Example 1 Find the Surface Area and the Volume of a sphere with a radius of 2. 2 November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Your Turn Find the surface area and volume. November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 1 A snowman is made with three spheres. The largest has a diameter of 24 inches, the next largest has a diameter of 20 inches, and the smallest has a diameter of 16 inches. Find the volume of the snowman. November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 1 Solution A snowman is made with three spheres. The largest has a diameter of 24 inches, the next largest has a diameter of 20 inches, and the smallest has a diameter of 16 inches. Find the volume of the snowman. The radii are 12 in, 10 in, and 8 in. November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 1 Solution The radii are 12 in, 10 in, and 8 in. 13,571.68 cu in November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Or, for easier calculation… November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 2 This is a grain silo, as found on many farms. They are used to store feed grain and other materials. They are usually cylindrical with a hemispherical top. Assume that the concrete part has a height of 50 feet, and the diameter of the cylinder is 18 feet. Find the volume of the silo. November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 2 Solution Volume of Cylinder V = r2h V = (92)(50) V =   81  50 V = 4050 V  12723.5 cu. ft. 18 50 9 November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 2 Solution Volume of Hemisphere 18 50 9 This is the volume of a sphere. The volume of the hemisphere is half of this value, which is 1526.8 cu. ft. November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 2 Solution Volume of Cylinder 12723.5 Volume of Hemisphere 1526.8 Total Volume 12723.5 + 1526.8 = 14250.3 cu. ft. 18 50 9 November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 2 Extension Total Volume =14250.3 cu. ft. One bushel contains 1.244 cubic feet. How many bushels are in the silo? 14250.3  1.244 = 11455.2 bushels November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 3 A sphere is inscribed inside a cube which measures 6 in. on a side. What is the ratio of the volume of the sphere to the volume of the cube? November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 3 Solution Volume of the Cube: 6  6  6 = 216 cu in Radius of the Sphere: 3 in. Volume of the Sphere: 6 6 3 6 November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Problem 3 Solution 6 6 6 Volume of the Cube: 216 cu in Volume of the Sphere: 36 Ratio of Volume of Sphere to Volume of Cube 6 6 3 6 November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 4 A mad scientist makes a potion in a full spherical flask which has a diameter of 4 inches. To drink it, he pours it into a cup with a diameter of 3.5 inches and is 3.5 inches high. Will the potion fit into the cup? If not, how much is left in the flask? November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 4 Solution Flask Volume: Diameter = 4 inches Radius = 2 inches November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 4 Solution 33.5 cu in Cup Volume: Diameter = 3.5 inches Radius = 1.75 inches Height = 3.5 inches November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 4 Solution 33.5 cu in 33.7 cu in The flask holds 33.5 cu in. The cup holds 33.7 cu in. Yes, the potion fits into the cup. November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 5 The surface area of a sphere is 300 cm2. Find: Its radius, The circumference of a great circle, Its volume. November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Problem 5 Solutions November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Last Problem On a far off planet, Zenu was examining his next target, Earth. The radius of the Earth is 3963 miles. What is the volume of material that will be blown into space? November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Last Problem Solution November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Summary A sphere is the set of points in space equidistant from a center. A hemisphere is half of a sphere. A great circle is the largest circle that can be drawn on a sphere. The diameter of a great circle equals the diameter of the sphere. November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Formulas to Know r November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres

Geometry 11.7 Surface Area and Volume of Spheres Homework Page 762 Problems 10 – 17, 20 – 29 all November 13, 2018 Geometry 11.7 Surface Area and Volume of Spheres