Sec. 11 – 4 Volumes of Prisms & Cylinders Objectives: 1) To find the volume of a prism. 2) To find the volume of a cylinder.

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Sec. 11 – 4 Volumes of Prisms & Cylinders Objectives: 1) To find the volume of a prism. 2) To find the volume of a cylinder.

Volume ►V►V►V►Volume – Is the space that a figure occupies. MMMMeasured in cubic units. ►c►c►c►cm3, in3, m3, ft3 ►T►T►T►Th(10-5) Cavalieri’s Principle IIIIf 2 space figures have the same height & the same cross-sectional every level, then they have the same volume.

I. Finding the volume of a Prism ► Prism – 2  parallel bases, sides (Lats) are rectangles. Height (h) Area of Base (B) V = Bh Height of Prism Area of Base A = bh (Rectangle) A = ½bh (Triangle) A = ½ap (Polygon)

Ex. 1: Finding the Volume of a rectangular prism ► The box shown is 5 units long, 3 units wide, and 4 units high. How many unit cubes will fit in the box? What is the volume of the box?

Ex.1: Find the Volume of the Prism 5in 3in 10in V = Bh = (3in 5in)(10in) = (15in 2 )(10in) = 150in 3 Area of Base B = lw

Ex.2: Find the volume of the following 20m 29m 40m V = Bh = ½bh h = ½(20m)__ (40m) = 210m 2 40m = 8400m 3 Height of the base: a a 2 + b 2 = c 2 a = 29 2 b = 21 Triangle 21

Ex.3: Yet another prism! Find the volume. 8in 10in V = Bh = ½bh h = ½(8in) __ (10in) = (27.7in 2 ) (10in) = 277in h 60° Sin 60 = h/8.866 = h/8 6.9 = h 6.9

Finding Volumes of prisms and cylinders. ► Bonaventura Cavalieri ( ). To see how it can be applied, consider the solids on the next slide. 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.

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.

II. Volume of a Cylinder r h V = Bh Volume of right cylinder Height of cylinder Area of base: (Circle) A =  r 2

Ex.4: Find the area of the following right cylinder. 16ft 9ft V = Bh =  r 2 h =  (8ft) 2 (9ft) = 64  ft 2 (9ft) = 576  ft 3 = ft 3 Area of a Circle

Ex.5: Find the volume of the following composite figure. Half of a cylinder: V c = Bh =  r 2 h =  (6in) 2 (4in) = 452in 3 = 452/2 = 226in 3 12in 4in 11in Volume of Prism: V p = Bh = (11)(12)(4) = 528in 3 V T = V c + V p = 226in in 3 = 754in 3

What have we learned?? Volume of a prism or a cylinder: V = Bh Capitol “B” stands for area of the base. Composite Figures: Made up of two separate solids.