12.4 Volume of Prisms & Cylinders Unit V Day 6

Do Now Find the area of each figure. a square with sides of length 4 a circle with a radius of 7 cm an equilateral triangle with sides of length 4 meters a regular hexagon with sides of length 2 feet 16 units2 49π, or about 153.9 cm2 4√3 , or about 6.93 m2 6√3 , or about 10.4 ft2

Exploring Volume The volume of a solid is the number of cubic units contained in its interior. Volume is measured in cubic units, such as cubic meters (m3).

Volume Postulates

Volume Theorems

Thm. 12.6: Cavalieri’s Principle Theorem 12.6 is named after Bonaventura Cavalieri (1598-1647). If two solids have the same height and the same cross-sectional area at every level, then they have the same volume. To see how it can be applied, consider the solids above. All three have cross sections with equal areas, B, and all three have equal heights, h. By Cavalieri’s Principle, it follows that each solid has the same volume. The point of this is that there is not a different volume formula for oblique prisms.

Ex. 1: Using Cavalieri’s Principle Use Cavalieri’s Principle to find the volume of the oblique cylinder. Even thought it’s oblique, volume is still V = Bh = πr2h V = π52 8 = 200π = 628.319 ft3

Ex. 2: Finding Volumes Find the volume of the right solids. A = ½ bh = ½ (3)(4) = 6 cm2 ; V = Bh = (6)(2) = 12 cm3 A = πr2 = π82 = 64π in.2 ; V = Bh = 64π(6) = 384πin.3 ≈ 1206.37 in.3

Ex. 3: Using Volumes Use the measurements given to solve for x. A side length of the cube is x feet. V = s3  100 = x3  4.64 ≈ x. So the height, width, and length of the cube are about 4.64 feet. b) The area of the base is πx2 meters squared. V = Bh. 4561 = πx2(12)  4561/(12π) = x2  11≈ x. So the radius of the cylinder is about 11 meters

Ex. 4: Using Volumes in Real Life (Construction) Concrete weighs 145 pounds per cubic foot. Find the volume of the concrete block shown. Find the weight of the concrete block shown. B = area of large rectangle – 2 * area of small rectangle B =(1.31)(0.66) – 2(0.33)(0.39) B ≈ 0.61 ft2 V = Bh V ≈ 0.61(0.66) = 0.40 ft.3 To find the weight of the block, multiply the pounds per cubic foot, 145 lb/ft.3 by the number of cubic feet, 0.40 ft3. Weight = 145 lb/1 ft3 * 0.40 ft3 ≈ 58 lbs.

Closure How do you find the volume of a prism or cylinder with height h? Multiply h by the area of a base B. In the case of a cylinder with a base radius of r, B = πr2.