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9.3 Cylinders & Cones

9.3 Cylinders & Cones. Cylinders. A cylinder has 2 bases that are congruent circles lying on parallel planes. The cylinder is a right cylinder if the line joining the centers of the circles called an axis is an altitude h —that is, perpendicular to the planes of the circular bases .

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9.3 Cylinders & Cones

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  1. 9.3Cylinders & Cones Section 9.3 Nack/Jones

  2. Cylinders • A cylinder has 2 bases that are congruent circles lying on parallel planes. • The cylinder is a right cylinder if the line joining the centers of the circles called an axis is an altitude h—that is, perpendicular to the planes of the circular bases. Section 9.3 Nack/Jones

  3. Surface Area of a Cylinder • Theorem 9.3.1: The lateral area L of a right circular cylinder with altitude of length h and circumference C of the base is given by: L = hC or L = 2rh • Theorem 9.3.2: The total area T of a right circular cylinder with base area B and lateral area L is given by: T = L + 2B or T = 2rh + 2r² Example: For the figure: L = 2 512 = 120 sq. units T = 120+ 25² = l70  sq. units. Ex. 1 p. 426 5 h = 12 Section 9.3 Nack/Jones

  4. Volume of a Cylinder • Theorem 9.3.3: The volume V of a right circular cylinder with base area B and altitude of length h is given by: V = Bh or V = r²h Example: The volume of the right cylinder with radius = 5 and height = 12 is: V =  5² (12) = 60  units3 Ex. 2,3 p. 426 Section 9.3 Nack/Jones

  5. Cone • The solid figure formed by connecting a circle with a point not in the plane of the circle is called a cone. • A cone has one base. • Right cone: the altitude passes through the center of the base circle. Also the slant height is the distance l. • Oblique cone: If the altitude, h, is not perpendicular to the base. Section 9.3 Nack/Jones

  6. Surface Area of a Cone • Theorem 9.3.4: The lateral area L of a right circular cone with slant height of length l and circumference C of the base is given by: L = ½l C or L = ½l (2r) • Theorem 9.3.5: The total area T of a right circular cone with base area B and lateral area L is given by: T = B + L or T = r² + rl Ex 4 p.428 • Theorem 9.3.6: In a right circular cone, the lengths of the radius r (of the base), the altitude h, and the slant height l satisfy the Pythagorean Theorem; that is l² = r² + h² in every right circular cone. Section 9.3 Nack/Jones

  7. Volume of a Cone • Theorem 9.3.7: The volume V of a right circular cone with base area B and altitude of length h is given by: V = ⅓ Bh or V = ⅓ r²h Table 9.3 p. 429 Note: The volume of an oblique and right circular cone are computed with the same formula. Fig. 9.37 p. 430 Solids of Revolution • Locus of points when we rotate a plane region around a line segment which becomes the axis of the resulting solid formed. Figure 9.35 and Example 5,6 p. 429 Section 9.3 Nack/Jones

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