Objectives Find volumes of prisms. Find volumes of cylinders.
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.
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 Remember, B = Area of Base. h
Find the volume of the right triangular prism. Example 1: Volume of a Triangular Prism 24 15 a 20 Use the Pythagorean Theorem to find the length of the base of the prism. *Note: Remember, you can only use P.T. on Right Triangles!
Using Pythagorean Theorem to find the length of the base of the prism we get... Example 1: Volume of a Triangular Prism 24 20 15 a a² + b² = c² Pythagorean Theorem a² + 15² = 24² a² + 225 = 576 a² = 351 a = √351 a ≈ 18.7
Example 1: Volume of a Triangular Prism 24 20 15 18.7 V = Bh Volume of a Prism So… V =½(18.7)(15)(20) B = ½(18.7)(15) h = 20 V = 2,805 cubic centimeters
Find the volume in feet of the rectangular prism. Example 2: Volume of a Rectangular Prism First, we must convert inches to feet. 12 in.25 ft. 10 ft 12 inches = 1 foot
Now, we can find the volume in feet of the rectangular prism. Example 2: Volume of a Rectangular Prism 1 ft.25 ft. 10 ft. 1 ft. x 10 ft. x 25 = 250 ft.
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 Area of Base = πr² h r
Find the volume of each cylinder. Example 3: Volume of a Cylinder 9.4m 1.6m The height h is 9.4 meters, and the radius r is 1.6 meters. V = πr²h = π(1.6²)(9.4) ≈ 75.6 meters
Find the volume of each cylinder. 7 in. 15 in. 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. a² + b² = c² Pythagorean Theorem h² + 7² = 15² h² + 49 = 225 h² = 176 h ≈ 13.3 Example 4: Volume of a Cylinder
Find the volume of each cylinder. 7 in. 13.3 in. V = π(3.5²)(13.3) V = 511.8 The volume is approximately 511.8 cubic inches. Example 4: Volume of a Cylinder
Principle 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
Find the volume of the oblique cylinder. Example 5: Volume of an Oblique Cylinder 8 yd 13 yd To find the volume, use the formula for a right cylinder. V = πr²h = π(8²)(13) = 2,613.8 The volume is approximately 2,613.8 cubic yards.