11.5 Volume of Prisms and Cylinders Geometry

Geometry 11.5 Volume of Prisms and Cylinders Goals Find the volume of prisms. Find the volume of cylinders. Solve problems using volume. Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders The number of cubic units contained in a solid. Measured in cubic units. Basic Formula: V = Bh B = area of the base, h = height Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Cubic Unit V = s3 V = 1 cu. unit s 1 1 s 1 s Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Which stack has the largest Volume? Cavalieri’s Principle If two solids have the same height and the same cross-sectional area at every level, then they have the same volume. Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Prism: V = Bh B B h h h B Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Cylinder: V = r2h r B h h V = Bh Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Example 1 Find the volume. Triangular Prism V = Bh Base = 40 V = 40(3) = 120 10 8 3 Abase = ½ (10)(8) = 40 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Example 2 Find the volume. V = Bh The base is a ? Hexagon 12 10 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Example 2 Solution 12 12 12 ? ? ? 6 10 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Example 2 Solution V = Bh V = (374.1)(10) V  3741 12 374.1 10 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Example 3 A soda can measures 4.5 inches high and the diameter is 2.5 inches. Find the approximate volume. V = r2h V = (1.252)(4.5) V  22 in3 (The diameter is 2.5 in. The radius is 2.5 ÷ 2 inches.) Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Example 4 A wedding cake has three layers. The top cake has a diameter of 8 inches, and is 3 inches deep. The middle cake is 12 inches in diameter, and is 4 inches deep. The bottom cake is 14 inches in diameter and is 6 inches deep. Find the volume of the entire cake, ignoring the icing. Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Example 4 Solution VTop = (42)(3) = 48  150.8 in3 VMid = (62)(4) = 144  452.4 in3 VBot = (72)(6) = 294  923.6 in3 r = 4 8 3 r = 6 12 4 486  1526.8 in3 14 6 r = 7 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Concrete Pipe Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Example 5 A manufacturer of concrete sewer pipe makes a pipe segment that has an outside diameter (o.d.) of 48 inches, an inside diameter (i.d.) of 44 inches, and a length of 52 inches. Determine the volume of concrete needed to make one pipe segment. 48 44 52 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Example 5 Solution Strategy: Find the area of the ring at the top, which is the area of the base, B, and multiply by the height. View of the Base Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Example 5 Solution Strategy: Find the area of the ring at the top, which is the area of the base, B, and multiply by the height. Area of Outer Circle: Aout = (242) = 576 Area of Inner Circle: Ain = (222) = 484 Area of Base (Ring): ABase = 576 - 484 = 92 48 44 52 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Example 5 Solution V = Bh ABase = B = 92 V = (92)(52) V = 4784 V  15,029.4 in3 48 44 52 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Example 5 Alternate Solution 48 Vouter = (242)(52) Vouter = 94,096.98 Vinner = (222)(52) Vinner = 79,067.60 V = Vouter – Vinner V  15,029.4 in3 44 52 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Example 6 4 5 L A metal bar has a volume of 2400 cm3. The sides of the base measure 4 cm by 5 cm. Determine the length of the bar. Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Example 6 Solution 4 5 L Method 1 V = Bh B = 4  5 = 20 2400 = 20h h = 120 cm Method 2 V = L  W  H 2400 = L  4  5 2400 = 20L L = 120 cm Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Example 7 Diameter = 7 3 in V = 115 in3 A 3-inch tall can has a volume of 115 cubic inches. Find the diameter of the can. Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Summary The volumes of prisms and cylinders are essentially the same: V = Bh & V = r2h where B is the area of the base, h is the height of the prism or cylinder. Use what you already know about area of polygons and circles for B. Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders B h h V = r2h V = Bh Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Which Holds More? This one! 3.2 in 1.6 in 4 in 4.5 in 2.3 in Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders What would the height of cylinder 2 have to be to have the same volume as cylinder 1? r = 3 r = 4 #2 #1 8 h Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Solution r = 4 #1 8 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Geometry 11.5 Volume of Prisms and Cylinders Solution r = 3 #2 h Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

Similar Solids We learned that the ratio of the areas is the square of the ratio of the sides in lesson 8.1. 5 125 ? 8 320 A= 320 Find Area of smaller polygon February 16, 2019 Geometry 8.1 Similar Polygons

Geometry 11.5 Volume of Prisms and Cylinders Similar Solids The ratio of the volume of similar figures is the cube of the ratio of sides. The solids are similar. Find the volume of solid B       V = 13564.8 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders