Volume of a Cylinder Lesson 1

Volume of a Cylinder Words: 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 = πr2 or V = πr2h 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 = πr2

The volume is about 651.9 cubic centimeters. Example 1 Find the volume of the cylinder. Round to the nearest tenth. V =πr2h V = π(5)2(8.3) Use a calculator. V = 651.8804756 The volume is about 651.9 cubic centimeters.

If the diameter is 16, the radius is 8. Example 2 Find 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 =πr2h V = π(8)2(20) Use a calculator. V = 4,021.2 The volume is about 4,021.2 cubic inches.

Got it? 1 & 2 Find 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

The paperweight weighs about 8.46 ounces. Example 3 A 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 =πr2h V = π(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.

The container weighs about 261.5 ounces. Got it? 3 The 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 =πr2h V = π(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 Prism V =lwh V = (12)(12)(12) V = 1,728 Cylinder V =πr2h V = π(1)2(12) V = 37.7 The volume of the bead is 1,728 – 37.7 or 1,690.3 cubic millimeters.

Half of the volume is (0.5)(24,543.6926) Got it? 4 The 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 Prism V =lwh V = (50)(25)(30) V = 37,500 Half Cylinder V =πr2h V = π(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.

Volume of Cones Lesson 2

Volume of a Cone A cone is a three-dimensional figure with one circular base. Words: the volume V of a cone with radius r is one third the area of the base B times the height h. Symbol: 𝑉= 1 3 𝐵ℎ 𝑜𝑟 𝑉= 1 3 𝜋𝑟2ℎ Model:

The volume of the cone is about 56.5 cubic inches. Example 1 Find the volume of the cone. Round to the nearest tenth. V = 𝟏 𝟑 πr2h V = 𝟏 𝟑 π(3)2(6) V ≈ 56.5 The volume of the cone is about 56.5 cubic inches.

Got it? 1 Find the volume of each cone. Round to the nearest tenth. a. b. V = 𝟏 𝟑 πr2h V = 𝟏 𝟑 π(2)2(7) V ≈ 29.3 The volume of the cone is about 29.3 cubic feet. V = 𝟏 𝟑 πr2h V = 𝟏 𝟑 π(12)2(26) V ≈ 3,920.7 The volume of the cone is about 3,920.7 cubic feet.

The volume of the paper cup is about 167.6 cubic centimeters. Example 2 A 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. V = 𝟏 𝟑 πr2h V = 𝟏 𝟑 π(4)2(10) V ≈ 167.6 The volume of the paper cup is about 167.6 cubic centimeters.

The total volume is 21.2(6) or 127.2 cubic inches. Got it? 2 April is filling six identical cones for her piñata. 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. V = 𝟏 𝟑 πr2h V = 𝟏 𝟑 π(1.5)2(9) V ≈ 21.2 The total volume is 21.2(6) or 127.2 cubic inches.

Example 3 Find the volume of the solid. Round to the nearest tenth. Volume of Cylinder: V =πr2h V = π(4)2(4) V ≈ 201.1 Find the volume of the solid. Round to the nearest tenth. Volume of Cone: V = 𝟏 𝟑 πr2h V = 𝟏 𝟑 π(4)2(5) V ≈ 83.8 The total volume is 201.1 + 83.8 or 284.9 cubic feet.

The total volume is 697.4 inches3. Got it? 3 Find the volume of the solid. Round to the nearest tenth. Volume of 2nd Cone: V = 𝟏 𝟑 πr2h V = 𝟏 𝟑 π(6)2(13) Volume of 1st Cone: V = 𝟏 𝟑 πr2h V = 𝟏 𝟑 π(6)2(5.5) The total volume is 697.4 inches3.

Volume of Spheres Lesson 3

Real-World Link Brittani purchased a necklace that contained a round pearl with a diameter of 7.5 millimeters. What is the circumference of the largest circle around the outside of the pearl? Round to the nearest tenth. 𝐶=2𝜋𝑟 𝐶=2𝜋 3.75 𝐶=23.6

Volume of a Sphere Words: the volume V of a sphere is four thirds the product of π and the cube of the radius r. Symbol: 𝑉= 4 3 π𝑟3 Model:

The volume is about 904.8 cubic millimeters. Example 1 Find the volume of this sphere. Volume of Sphere: V = 𝟒 𝟑 πr3 V = 𝟒 𝟑 π(6)3 V ≈𝟗𝟎𝟒.𝟖 The volume is about 904.8 cubic millimeters.

Got it? 1 Find the volume of each sphere. a. b. V = 𝟒 𝟑 πr3 V ≈ 5,575.8 cm3 V = 𝟒 𝟑 πr3 V = 𝟒 𝟑 π(5)3 V ≈ 523.6 mm3

The stone is about 268.1 cubic feet. Example 2 A 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. V = 𝟒 𝟑 πr3 V = 𝟒 𝟑 π(4)3 V ≈ 268.1 feet3 The stone is about 268.1 cubic feet.

The stone is about 7.24 cubic inches. Got it? 2 A 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. V = 𝟒 𝟑 πr3 V = 𝟒 𝟑 π(1.2)3 V ≈ 7.24 inches3 The stone is about 7.24 cubic inches.

It would take 1.6 minutes to inflate the ball. Example 3 A 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. V = 𝟒 𝟑 πr3 V = 𝟒 𝟑 π(5)3 V ≈ 523.6 inches3 𝟓𝟐𝟑.𝟔 𝟑𝟐𝟓 =𝟏.𝟔 It would take 1.6 minutes to inflate the ball.

It would take 62.8 minutes to inflate the ball. Got it? 3 A 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. V = 𝟒 𝟑 πr3 V = 𝟒 𝟑 π(3)3 V ≈ 113.09733 cm3 𝟏𝟏𝟑.𝟎𝟗𝟕 𝟏.𝟖 =𝟔𝟐.𝟖𝟑𝟏 It would take 62.8 minutes to inflate the ball.

The volume is about 261.8 cubic centimeters. Volume of a Hemisphere Example 4: Find the volume of the hemisphere. Round to the nearest tenth. V =( 𝟏 𝟐 ) 𝟒 𝟑 πr3 V =( 𝟏 𝟐 ) 𝟒 𝟑 π(5)3 V ≈ 261.8 cm3 The volume is about 261.8 cubic centimeters.

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.

Surface Area of Cylinders Lesson 4

Lateral Surface Area Words: 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. = 2πrh

Total Surface Area Words: The surface area (S.A) is the lateral surface area plus the area of the two circular bases. Symbols: S.A. = L.A. + 2πr2 or S.A. = 2πrh + 2πr2

Finding the surface area with a net: What’s the difference between lateral and total surface area?

The surface area is about 113.1 square meters. Example 1 Find the surface area of the cylinder. Round to the nearest tenth. S.A. = 2πrh + 2πr2 S.A. = 2π(2)(7) + 2π(2)2 S.A. ≈ 113.1 The surface area is about 113.1 square meters.

Got it? 1 Find the surface area of each cylinder. Round to the nearest tenth. a. b. S.A. = 2πrh + 2πr2 S.A. = 2π(3)(9) + 2π(3)2 S.A. ≈ 226.2 The surface area is about 113.1 square feet. S.A. = 2πrh + 2πr2 S.A. = 2π(7)(10) + 2π(7)2 S.A. ≈ 747.7 The surface area is about 747.7 square centimeters.

Example 2 A circular fence that is 2 feet high is to be built around the outside of a carousel. The distance from the center of the carousel to the edge of the fence will be 35 feet. What is the area of the fencing material that is needed to make the fence around the carousel? You will need to find the lateral surface area. Radius is 35 and height is 2. L.A. = 2πrh L.A. = 2π(35)(2) L.A. ≈ 439.8 About 440 square feet of material is needed to make the fence.

Got it? 2 a. 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. L.A. = 2πrh L.A. = 2π(5.1)(2.9) L.A. ≈ 92.9cm2 The lateral surface area is about 92.9 square centimeters. S.A. = 2πrh + 2πr2 S.A. = 2π(2)(2.5) + 2π(2)2 S.A. ≈ 56.5in2 The surface area is about 56.5 square inches.

Surface Area of Cones Lesson 5

The slant height is about 6.3 inches. Real-World Link Bobby is making waffle cones from scratch. Use the Pythagorean Theorem to find the slant height ℓ of the cone if the radius is 2 inches and the height is 6 inches. a2 + b2 = c2 22 + 62 = 𝓵2 40 = 𝓵2 The slant height is about 6.3 inches.

Lateral Surface Area of a Cone Words: The lateral surface area (L.A) of a cone is π times the radius times the slant height ℓ. Symbols: L.A. = πrℓ Model:

The lateral surface area is about 204.2 square millimeters. Example 1 Find the lateral area of the cone. Round to the nearest tenth. L.A. = πrℓ L.A. = π(𝟓)(𝟏𝟑) L.A. ≈ 204.2 The lateral surface area is about 204.2 square millimeters.

Got it? 1 a. 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. L.A. = πrℓ L.A. = π(𝟒)(𝟗.𝟓) L.A. ≈ 119.4 in2 The lateral surface area is about 119.4 square inches. L.A. = πrℓ L.A. = π(𝟖)(𝟏𝟎) L.A. ≈ 251.3 cm2 The lateral surface area is about 252.3 square centimeters.

Total Surface Area of a Cone Words: The surface area (S.A) of a cone is the lateral surface area plus the area of the base. Symbols: S.A. = πrℓ + πr2 Model:

The surface area is about 230.0 square inches. Example 2 Find the surface area of the cone. Round to the nearest tenth. S.A. = πrℓ + πr2 S.A. = π(𝟔)(6.2) + π(6)2 S.A. ≈ 230.0 in2 The surface area is about 230.0 square inches.

The surface area is about 346.4 square millimeters. Got it? 2 Find the surface area of the cone. Round to the nearest tenth. S.A. = πrℓ + πr2 S.A. = π(𝟒.𝟓)(20) + π(4.5)2 S.A. ≈ 346.4 mm2 The surface area is about 346.4 square millimeters.

The amount of fabric used is about 118.5 square feet. Example 3 A tepee has a radius of 5 feet and a slant height of 12 feet. What formula would you use to find the area of the fabric? Lateral Surface Area Find the lateral surface area. L.A. = πrℓ L.A. = π(𝟓)(𝟏𝟐) L.A. ≈ 118.5 ft2 The amount of fabric used is about 118.5 square feet.

The amount of fabric used is about 138.2 square inches. Got it? 3 Rachel bought a party hats that were in the shape of a cone. Each hat has a diameter of 8 inches and a slant height of 11 inches. What formula would you use to find the area of the hat? Lateral Surface Area Find the lateral surface area. L.A. = πrℓ L.A. = π(𝟒)(𝟏𝟏) L.A. ≈ 138.2 in2 The amount of fabric used is about 138.2 square inches.

Ticket Out The Door Write the formula for the lateral surface area. Explain what each variable means. Use complete sentences. (2-3 sentences)

Change in Dimensions Lesson 6

Real-World Link Stephen is creating a model of the Washington Monument for history class. The model will be 1 100 of the monument’s actual size. The square pyramid that sits atop the monument’s obelisk shape has a slant height of about 57.6 feet. Each side of the pyramid’s base is about 34 feet. What is the are of the of the triangular faces of the actual pyramid? A = 𝟏 𝟐 bh A = 𝟏 𝟐 (34)(57.6) A = 979.2 feet2

Real-World Link Stephen is creating a model of the Washington Monument for history class. The model will be 1 100 of the monument’s actual size. The square pyramid that sits atop the monument’s obelisk shape has a slant height of about 57.6 feet. Each side of the pyramid’s base is about 34 feet. What is the slant height of the pyramid of the model Stephen is creating? Model Slant Height = 𝟏 𝟏𝟎𝟎 x Actual Slant Height Model Slant Height = 𝟏 𝟏𝟎𝟎 (57.6) Model Slant Height = 0.567 feet

Real-World Link Stephen is creating a model of the Washington Monument for history class. The model will be 1 100 of the monument’s actual size. The square pyramid that sits atop the monument’s obelisk shape has a slant height of about 57.6 feet. Each side of the pyramid’s base is about 34 feet. What is the length of one base of the pyramid on the model? Model Base Length= 𝟏 𝟏𝟎𝟎 x Actual Base Length Model Base Length = 𝟏 𝟏𝟎𝟎 (34) Model Base Length = 0.34 feet

Real-World Link Stephen is creating a model of the Washington Monument for history class. The model will be 1 100 of the monument’s actual size. The square pyramid that sits atop the monument’s obelisk shape has a slant height of about 57.6 feet. Each side of the pyramid’s base is about 34 feet. What is the area of one of the triangular faces of the model pyramid? A = 𝟏 𝟐 (Base)(Height) A = 𝟏 𝟐 (0.34)(0.576) Area = 0.09792 ft2

𝒎𝒐𝒅𝒆𝒍 𝒂𝒓𝒆𝒂 𝒂𝒄𝒕𝒖𝒂𝒍 𝒂𝒓𝒆𝒂 = 𝟎.𝟎𝟗𝟕𝟗𝟐 𝟗𝟕𝟗𝟐 = 𝟏 𝟏𝟎,𝟎𝟎𝟎 Real-World Link Write a ratio comparing the area of the triangular side of the model to the actual monument. 𝒎𝒐𝒅𝒆𝒍 𝒂𝒓𝒆𝒂 𝒂𝒄𝒕𝒖𝒂𝒍 𝒂𝒓𝒆𝒂 = 𝟎.𝟎𝟗𝟕𝟗𝟐 𝟗𝟕𝟗𝟐 = 𝟏 𝟏𝟎,𝟎𝟎𝟎 The actual area of the monument is 10,000 times as large as the scale model.

Surface Area of Similar Solids Solids 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 it’s area.

Example 1 The 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 32 S.A. = 78 x 9 S.A. = 702 cm2 The surface area of the larger rectangular prism is 702 squared centimeters.

Got it? 1 a. The surface area of a triangular prism is 34 square inches. What is the surface area of a similar prims that is twice as large? S.A. = 136 in2 b. The world’s largest box of raisins has a surface area of 352 square feet. If a similar box is smaller than the largest box by a scale factor of 1 48 , what is its surface area? S.A. = 352 x 𝟏 𝟒𝟖 𝟐 S.A. = 352 x 𝟏 𝟐𝟑𝟎𝟒 S.A. ≈ 0.15 feet2

Volume of Similar Solids Solids 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 it’s volume.

Example 2 A triangular prims has a volume of 432 cubic yards. If the prism is reduced to one third its original size, what is the volume of the new prism? V = 432 x ( 𝟏 𝟑 )3 V = 432 x 𝟏 𝟐𝟕 V = 16 yards3 The new prism will have a volume of 16 cubic yards.

Got it? 2 a. 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 3 The 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.