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13.1 Volumes of Prisms and Cylinders

By Nathan Critchfield and Ben Tidwell. 13.1 Volumes of Prisms and Cylinders. Objectives. Find volumes of prisms. Find volumes of cylinders. Volumes of Prisms.

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13.1 Volumes of Prisms and Cylinders

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  1. By Nathan Critchfield and Ben Tidwell 13.1 Volumes of Prisms and Cylinders

  2. Objectives • Find volumes of prisms. • Find volumes of cylinders.

  3. Volumes of Prisms • The volume of a figure is the measure of the amount of space that a figure encloses. Volume is measured in cubic units. You can create a rectangular prism from different views of the figure to investigate its volume.

  4. Volumes of Prisms • If a prism has a volume of V cubic units, a height of h units, and each base has an area of B square units, then V = Bh Area of base = B h

  5. Example 1: Volume of a Triangular Prism Find the volume of the right triangular prism Use the Pythagorean Theorem to find the length of the base of the prism. 24 *Note: Remember, you can only use P.T. on Right Triangles! 15 20 a

  6. Example 1: Volume of a Triangular Prism Use Pythagorean Theorem to find the length of the base of the prism. a² + b² = c²  Pythagorean Theorem a² + 15² = 24² 24 a² + 225 = 576 a² = 351 15 a = √351 20 a ≈ 18.7 a

  7. Example 1: Volume of a Triangular Prism Find the volume of the prism. V = Bh  Volume of a Prism 24 B = 18.7(15) h = 20 BUT WAIT!.. Since it is a triangle, not a rectangle, it is… V =½(18.7)(15)(20) 15 20 V = 2,805 cubic centimeters 18.7

  8. Example 2: Volume of a Rectangular Prism Find the volume in feet of the rectangular prism Convert feet to inches. 10 ft 25 feet = 25 x 12 or 300 inches 12 in. 25 ft. 10 feet = 10 x 12 or 120 inches

  9. Example 2: Volume of a Rectangular Prism Find the volume in feet of a rectangular prism 300 in. x 120 in. = 36,000 in. 120 in. 36,000 in. x 12 in. = 432,000 cubic inches. 12 in. 300 in. 432,000 / 123 = 250 cubic feet.

  10. Volumes of Cylinders If a cylinder has a volume of V cubic units, a height of h units, and the bases have radii of r units, then V = Bh or V = πr²h h Area of base = πr² r

  11. Example 3: Volume of a Cylinder Find the volume of each cylinder a. The height h is 9.4 meters, and the radius r is 1.6 meters. V = πr²h 9.4m = π(1.6²)(9.4) ≈ 75.6 meters 1.6m

  12. Example 3: Volume of a Cylinder Find the volume of each cylinder b. a² + b² = c²  Pythagorean Theorem h² + 7² = 15² h² + 49 = 225 h² = 176 7 in. 15 in. h ≈ 13.3 The diameter of the base, the diagonal, and the lateral edge of the cylinder form a right triangle. Use the Pythagorean Theorem to find the height.

  13. Example 3: Volume of a cylinder Find the volume of each cylinder b. V = π(3.5²)(13.3) V = 511.8 7 in. 13.3 in. The volume is approximately 511.8 cubic inches.

  14. Key Concept! 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. • Which basically means, that whether it is right or oblique, it’s volume is V=Bh

  15. Example 4: Volume of an Oblique Cylinder Find the volume of the oblique cylinder 8 yd To find the volume, use the formula for a right cylinder. 13 yd V = πr²h = π(8²)(13) = 2,613.8 The volume is approximately 2,613.8 cubic yards.

  16. AssignmenT • Page 692 7-16, 20, 22-24

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