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=r2h

Volume Formulas Cone - V=1/3Bh

Cones Pringles A third of the calories 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?? Pringles A third of the calories

Volume Formulas Pyramid - V=1/3 Bh, where B is the area of the base and h is the height. Sphere - V=4/3r3 h

Example A = ½ bh Area of a triangle A = ½ (3)(4) Substitute values Find the volume of the right prism. A = ½ bh Area of a triangle A = ½ (3)(4) Substitute values A = 6 cm2 Multiply values -- base V = Bh Volume of a prism formula V = (6)(2) Substitute values V = 12 cm3 Multiply values & solve

Example A = r2 Area of a circle A = 82 Substitute values Find the volume of the right cylinder. A = r2 Area of a circle A = 82 Substitute values A = 64 in.2 Multiply values -- base V = Bh Volume of a prism formula V = 64(6) Substitute values V = 384in.3 Multiply values & solve V = 1206.37 in.3 Simplify

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

Example V = (⅓)Bh = (⅓)l•w•h = (⅓)15•15•22 = (⅓)4950 = 1650cm3 Find the volume of a square pyramid with base edges of 15cm & a height of 22cm. Square V = (⅓)Bh = (⅓)l•w•h = (⅓)15•15•22 = (⅓)4950 = 1650cm3 22cm 15cm 15cm

Example: Find the volume of the following right cone w/ a diameter of 6in. Circle V = ⅓Bh = (⅓)r2h = (⅓)(3)2(11) = (⅓)99 = 33 = 103.7in3 11in 3in

Example Recall: Ex. 5: If the volume of the cylinder is 441π m3, what is the volume of the cone? 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)r2h = (⅓)(8)2(10) = (⅓)(640) = 213.3 = 670.2cm3 10cm 4cm Volume of Cylinder NEXT! Vc = Bh = r2h = (8)2(4) = 256 = 804.2cm3 8cm VT = Vc + Vc VT = 670cm3 + 804.2cm3 VT = 1474.4cm3

Example V = ⅓Bh V = ⅓(r2)h 110 = (⅓)r2(10) 110 = (⅓)r2(10) The following cone has a volume of 110. What is its radius. V = ⅓Bh V = ⅓(r2)h 110 = (⅓)r2(10) 110 = (⅓)r2(10) 11 = (⅓)r2 33 = r2 r = √(33) = 5.7cm 10cm r

Example Find the volume of a sphere with a radius of 3 ft. V = 36 ft3 or 113.1 ft3

Find the radius of a sphere with a volume of 2304  cm3 Example Find the radius of a sphere with a volume of 2304  cm3