Volume and Surface Area

Lesson 1Volume of a CylinderVolume of a CylinderWords: The volume V of a cylinder with radius r is the area of the base B times the height h.

Symbols: V = Bh, where B = r2 or V = r2h

Model:

Volume the measure of space occupied by a solid

Cylinder a three-dimensional figure with two parallel congruent circular bases.

What is the formula do you use to find the area of the base of the cylinder?

A = r2

Example 1Find the volume of the cylinder. Round to the nearest tenth.

V =r2hV = (5)2(8.3)Use a calculator.V = 651.8804756

The volume is about 651.9 cubic centimeters. Example 2Find the volume of the cylinder with a diameter of 16 inches and a height of 20 inches. Round to the nearest tenth.

If the diameter is 16, the radius is 8. V =r2hV = (8)2(20)Use a calculator.V = 4,021.2

The volume is about 4,021.2 cubic inches. Got it? 1 & 2Find the volume of each cylinder. Round to the nearest tenth. a.

b. Diameter: 12 mm Height: 5 mm

V = 50.9 in.3 V = 565.4 mm3 Example 3A metal paperweight is in the shape of a cylinder. The paperweight has a height of 1.5 inches and a diameter of 2 inches. How much does the paperweight weigh if 1 square inch weighs 1.8 ounces? Round to the nearest tenth.

V =r2hV = (1)2(1.5)Use a calculator.V = 4.7

Multiply the volume by 1.8.4.7(1.8) = 8.46 The paperweight weighs about 8.46 ounces. Got it? 3The Roberts family uses a container shaped like a cylinder to recycle aluminum cans. It has a height of 4 feet and a diameter of 1.5 feet. The container is full. How much do the contents weigh if the average weight of the aluminum cans is 37 ounces per square foot? Round to the nearest tenth.

V =r2hV = (0.75)2(4)V = 7.068

Multiply the volume by 37.7.068(37) = 261.537588411 The container weighs about 261.5 ounces. Composite Shapes objects made up of more than one type of solid.Example 4:Tonya uses cube-shaped beads to make jewelry. Each bead has a circular hole through the middle. Find the volume of the bead.

Rectangular PrismV =lwhV = (12)(12)(12)V = 1,728

CylinderV =r2hV = (1)2(12)V = 37.7The volume of the bead is 1,728 37.7 or 1,690.3 cubic millimeters. Got it? 4The Service Club is building models of storage chests, like the one shown, to donate to charity. Find the volume of the chest to the nearest tenth.

Rectangular PrismV =lwhV = (50)(25)(30)V = 37,500Half CylinderV =r2hV = (12.5)2(50)V = 24,543.6926

Half of the volume is (0.5)(24,543.6926)The volume of the storage is 37,500 + 12,271.8463 = 49,771.8 centimeters cubed.

Lesson 2Volume of ConesVolume of a Cone

Example 1Find the volume of the cone. Round to the nearest tenth.

Got it? 1Find the volume of each cone. Round to the nearest tenth.a.

b.

Example 2A cone-shaped paper cup is filled with water. The height of the cup is 10 centimeters and the diameter is 8 centimeters. What is the volume of the paper cup? Round to the nearest tenth.

Got it? 2April is filling six identical cones for her piata. Each cone has a radius of 1.5 inches and a height of 9 inches. What is the total volume of the cones? Round the nearest tenth.Example 3Find the volume of the solid. Round to the nearest tenth.

The total volume is 201.1 + 83.8 or 284.9 cubic feet. Got it? 3Find the volume of the solid. Round to the nearest tenth.The total volume is 697.4 inches3.

Lesson 3Volume of Spheres

Real-World Link

Volume of a Sphere

Example 1Find the volume of this sphere.

Got it? 1Find the volume of each sphere. a.

b.

Example 2A spherical stone in the courtyard of the National Museum of Costa Rica has a diameter of about 8 feet. Find the volume of the spherical stone. Round to the nearest tenth.

Got it? 2A dish contains a spherical scoop of vanilla ice cream with a radius of 1.2 inches. What is the volume of the ice cream? Round to the nearest tenth.

Example 3A volleyball has a diameter of 10 inches. A pump can inflate the ball at a rate of 325 cubic inches per minute. How long will it take to inflate the ball? Round to the nearest tenth.

Got it? 3A snowball has a diameter of 6 centimeters. How long would it take the snowball to melt if it melts at a rate of 1.8 cubic centimeters per minute? Round to the nearest tenth.

Volume of a HemisphereExample 4:Find the volume of the hemisphere. Round to the nearest tenth.

Got it? 4 Find the volume of each hemisphere. Round to the nearest tenth.a.

b.

The volume is about 2.1 cubic centimeters. The volume is about 1,526.8 cubic meters.

Lesson 4Surface Area of CylindersLateral Surface AreaWords: The lateral area (L.A.) of a cylinder with height h and radius r is the circumference of the base times the height.

Symbols: L.A. = 2rh

Total Surface AreaWords: The surface area (S.A) is the lateral surface area plus the area of the two circular bases.

Symbols: S.A. = L.A. + 2r2 orS.A. = 2rh + 2r2

Finding the surface area with a net:

Whats the difference between lateral and total surface area? Example 1Find the surface area of the cylinder. Round to the nearest tenth.

Got it? 1Find the surface area of each cylinder. Round to the nearest tenth. a.

b.

Example 2Got it? 2a. Find the area of the label of a can of tuna with a radius of 5.1 centimeters and a height of 2.9 centimeters. Round to the nearest tenth.

b. Find the total surface areas of a cylindrical candle with a diameter of 4 inches and a height of 2.5 inches. Round to the nearest tenth.

Lesson 5Surface Area of ConesReal-World Link

Lateral Surface Area of a Cone

Example 1Find the lateral area of the cone. Round to the nearest tenth.

Got it? 1a. Find the lateral area of a cone with a radius of 4 inches and a slant height of 9.5 inches. Round to the nearest tenth.

b. Find the lateral area of a cone with a diameter of 16 centimeters and a slant height of 10 centimeters. Round to the nearest tenth. Total Surface Area of a Cone

Example 2Find the surface area of the cone. Round to the nearest tenth.

Got it? 2Find the surface area of the cone. Round to the nearest tenth.

Example 3Got it? 3Ticket Out The DoorWrite the formula for the lateral surface area. Explain what each variable means. Use complete sentences. (2-3 sentences)Lesson 6Change in DimensionsReal-World Link

Real-World LinkReal-World LinkReal-World LinkReal-World LinkWrite a ratio comparing the area of the triangular side of the model to the actual monument.

Surface Area of Similar SolidsSolids are similar if that have the same shape, and their measurements are proportional. The scale factor is 2. To find the surface area of the big cube, multiply the S.A. of the small cube by 4 or 22.2 because the scale factor is 2.Squared because its area. Example 1The surface area of a rectangular prism is 78 square centimeters. What is the surface area of a similar prims that is 3 times as large?S.A. = 78 x 32S.A. = 78 x 9S.A. = 702 cm2

The surface area of the larger rectangular prism is 702 squared centimeters. Got it? 1

Volume of Similar SolidsSolids are similar if that have the same shape, and their measurements are proportional. The scale factor is 2. To find the volume of the big cube, multiply the volume. of the small cube by 8 or 23.2 because the scale factor is 2.Cubed because its volume.

Example 2Got it? 2a. A square pyramid has a volume of 512 cubic centimeters. What is the volume of the square pyramid with dimensions one-forth of the original?V = 8 cm3

b. A cylinder has a volume of 432 cubic meters. What is the volume of a cylinder with dimensions one-third its size?

V = 16 meters3

Example 3The measurements for a standard hockey puck is shown. The giant hockey pock is 40 times the size of a standard puck. Find the volume and surface area of the giant puck. Surface Area and Volume of Original:

Surface Area and Volume of Original:

The volume of the giant hockey puck is 452,160 cubic inches, and the surface area is about 37,680 square inches.