10.4 Volumes of Prisms & Cylinders Geometry

Volume Volume – Is the space that a figure occupies. cm3, in3, m3, ft3 Measured in cubic units (it has 3 dimensions) cm3, in3, m3, ft3 The formulas for right prisms can also be used for oblique prisms

With rectangular prisms it is often easiest to use this formula: Volume = l ● w ● h For a cube you can use: Volume = s³

I. Finding the volume of a Prism Prism – 2 congruent parallel bases, sides 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? 5 ∙ 3 ∙ 4 = 60 u³

Ex.1: Find the Volume of the Prism

Ex.2: Find the volume of the following 29² - 20² = 441 441 = 21 ½ b ∙ h = ½ (20 ∙ 21) = ½ (420) = 210 B ∙ h = 210 ∙ 40 = 8400 m³ 29m a 40m 20m

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

B ∙ h = 200 m³ B ∙ 20 = 200 B = 10 B = ½ b ∙ h 10 = ½ 8 ∙ h 10 = 4 ∙ h Ex. 4 The volume of this figure is 200 m³. Find x. B ∙ h = 200 m³ B ∙ 20 = 200 B = 10 B = ½ b ∙ h 10 = ½ 8 ∙ h 10 = 4 ∙ h 2.5 = h = x X 20m 8m

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

Ex.5: 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.6: 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

Find the volume of the solid below. Rectangular prism: 25 ∙ 30 ∙ 25 = 18750 cm³ Cylinder hole: (r = 4 cm) Cylinder = π 4² ∙ 30 = 480π ≈ 1507.96 cm³ 18750 – 1508 ≈ 17242 cm³