Volumes of Prisms & Cylinders

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Volumes of Rectangular Prisms and Cylinders Lesson 9-9.

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Presentation transcript:

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

Volume Volume – Is the space that a figure occupies. Measured in cubic units. cm3, in3, m3, ft3 Th(10-5) Cavalieri’s Principle If 2 space figures have the same height & the same cross-sectional area @ every level, then they have the same volume.

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

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 Area of Base B = l•w V = Bh = (3in • 5in)(10in) = (15in2)(10in) = 150in3 10in 3in 5in

Ex.2: Find the volume of the following Triangle 29m V = Bh = ½bh • h = ½(20m)__ • (40m) = 210m2 • 40m = 8400m3 a 40m 21 20m Height of the base: a2 + b2 = c2 a2 + 202 = 292 b = 21

Ex.3: Yet another prism! Find the volume. 8 h 60° 4 10in Sin 60 = h/8 .866 = h/8 6.9 = h V = Bh = ½bh • h = ½(8in) __ • (10in) = (27.7in2) • (10in) = 277in3 8in 6.9

Finding Volumes of prisms and cylinders. Bonaventura Cavalieri (1598- 1647). 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.

V = Bh II. Volume of a Cylinder Height of cylinder r h Volume of right cylinder Area of base: (Circle) A = r2

V = Bh = r2 • h = (8ft)2 • (9ft) = 64ft2 • (9ft) = 576ft3 Ex.4: Find the area of the following right cylinder. Area of a Circle V = Bh = r2 • h = (8ft)2 • (9ft) = 64ft2 • (9ft) = 576ft3 = 1809.6ft3 16ft 9ft

Ex.5: Find the volume of the following composite figure. Half of a cylinder: Vc = Bh = r2•h = (6in)2 • (4in) = 452in3 = 452/2 = 226in3 11in 4in Volume of Prism: Vp = Bh = (11)(12)(4) = 528in3 12in VT = Vc + Vp = 226in3 + 528in3 = 754in3

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.