VOLUME OF CYLINDERS PYRAMIDS CONES AND SPHERES Volume

Volume • The volume of a solid is the number of cubic units contained

Finding Volumes • Cavalieri’s Principle is named after Bonaventura Cavalieri

Cavalieri’s Principle If two solids have the same height and the same cross-sectional area

Cavalieri’s Principle The six pieces maintain their same volume regardless of how they are

Volume Formulas • Prism - V=Bh, where B is the area of the base

Cones A fact: If Pringles came in a cone, which was the same height

Volume Formulas • Pyramid - V=1/3 Bh, where B is the area of the

Example • Find the volume of the right prism. A = ½ bh A

Example • Find the volume of the right cylinder. A = r 2 A

Example – Cavalieri’s • Find the volume if h = 10 and r =

Example • Find the volume of a square pyramid with base edges of 15

Example: Find the volume of the following right cone w/ a diameter of 6

Example Ex. 5: If the volume of the cylinder is 441π m 3, what

Ex. 4: Volume of a Composite Figure Volume of Cone first! Vc = ⅓Bh

Example • The following cone has a volume of 110. What is its radius.

Example Find the volume of a sphere with a radius of 3 ft. V

Example Find the radius of a sphere with a volume of 2304 cm 3

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VOLUME OF CYLINDERS, PYRAMIDS, CONES AND SPHERES

Volume • The volume of a solid is the number of cubic units contained in its interior.

Finding Volumes • Cavalieri’s Principle is named after Bonaventura Cavalieri

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.

Cavalieri’s Principle The six pieces maintain their same volume regardless of how they are moved

Volume Formulas • Prism - V=Bh, where B is the area of the base and h is the height. • Cylinder - V=Bh= r 2 h

Volume Formulas • Cone - V=1/3 Bh

Cones A fact: If Pringles came in a cone, which was the same height and diameter as the tall tube, it would contain one third of the calories!!! Why? ?

Volume Formulas • Pyramid - V=1/3 Bh, where B is the area of the base and h is the height. h • Sphere - V=4/3 r 3

Example • Find the volume of the right prism. A = ½ bh A = ½ (3)(4) A = 6 cm 2 V = Bh V = (6)(2) V = 12 cm 3 Area of a triangle Substitute values Multiply values -- base Volume of a prism formula Substitute values Multiply values & solve

Example • Find the volume of the right cylinder. A = r 2 A = 82 A = 64 in. 2 V = Bh V = 64 (6) V = 384 in. 3 V = 1206. 37 in. 3 Area of a circle Substitute values Multiply values -- base Volume of a prism formula Substitute values Multiply values & solve Simplify

Example – Cavalieri’s • Find the volume if h = 10 and r = 7

Example • Find the volume of a square pyramid with base edges of 15 cm & a height of 22 cm. Square V = (⅓)Bh = (⅓)l • w • h 22 cm 15 cm = (⅓)15 • 22 = (⅓)4950 = 1650 cm 3

Example: Find the volume of the following right cone w/ a diameter of 6 in. Circle 11 in 3 in V = ⅓Bh = (⅓) r 2 h = (⅓) (3)2(11) = (⅓)99 = 33 = 103. 7 in 3

Example Ex. 5: If the volume of the cylinder is 441π m 3, what is the volume of the cone? Recall: Ex. 6: If the radius of the cone in Ex. 5 is 7 m, what is its height?

Ex. 4: Volume of a Composite Figure Volume of Cone first! Vc = ⅓Bh = (⅓) r 2 h 10 cm = (⅓)(8)2 (10) = (⅓)(640) = 213. 3 = 670. 2 cm 3 4 cm 8 cm Volume of Cylinder NEXT! Vc = Bh VT = V c + V c = r 2 h VT = 670 cm 3 + 804. 2 cm 3 = (8)2(4) VT = 1474. 4 cm 3 = 256 = 804. 2 cm 3

Example • The following cone has a volume of 110. What is its radius. V = ⅓Bh V = ⅓( r 2)h 110 = (⅓) r 2(10) 10 cm 110 = (⅓)r 2(10) 11 = (⅓)r 2 r 33 = r 2 r = √(33) = 5. 7 cm

Example Find the volume of a sphere with a radius of 3 ft. V = 36 ft 3 or 113. 1 ft 3

Example Find the radius of a sphere with a volume of 2304 cm 3