12-3 Volume of Prisms & 12-3 Volume of Prisms & CylindersCylinders

p. 743p. 743

NCSCOS: 1.02, 2.03, 2.04NCSCOS: 1.02, 2.03, 2.04

Volume of a solidVolume of a solid

• DefnDefn – the amount of space inside the – the amount of space inside the solid.solid.

• Just think “how much water would it Just think “how much water would it take to fill the object”?take to fill the object”?

• Measured in cubic units (i.e. cmMeasured in cubic units (i.e. cm33))

Post. 27Post. 27 – Volume of a cube – Volume of a cube

V = sV = s33

V – volume & s – edge lengthV – volume & s – edge length

ExEx: find the volume of the cube.: find the volume of the cube.

V = sV = s33

V = 5V = 533

V = 125 mV = 125 m33

5 m5 m

Post. 28Post. 28 – Volume – Volume post. post.

• If 2 polyhedra are If 2 polyhedra are , then they , then they have the same volume.have the same volume.

Post. 29Post. 29 – Volume + post. – Volume + post.

• The volume of a solid is the sum of the The volume of a solid is the sum of the volumes of its non-overlapping parts.volumes of its non-overlapping parts.

Total Total volume volume volumevolume

volume = volume = of of + of + of

of tower pyramid prismof tower pyramid prism

Thm 12.6Thm 12.6 – Cavalieri’s Principle – Cavalieri’s Principle

• If 2 solids have the same height & the If 2 solids have the same height & the same cross-sectional area at every same cross-sectional area at every level, then they have the same volume.level, then they have the same volume.

• Basically, this means that right & Basically, this means that right & oblique solids use the same volume oblique solids use the same volume formulas, unlike surface area & lateral formulas, unlike surface area & lateral area formulas.area formulas.

Oblique Prisms & Cylinders

• Cavalieri’s Principle:

A right prism and an oblique prism with the same base and height have the same volume.

Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they have the same volume.

EXAMPLES: Find the volume of the oblique cylinder (exact value & nearest

tenth).

V = 5324 ≈ 16,725.8 cm3

Find the volume of the prism. Round to the nearest tenth if

necessary.

V = 203.7 ft3

Thm 12.7Thm 12.7 – Volume of a Prism – Volume of a PrismV = BhV = Bh

V – volume, B – area of the base, & V – volume, B – area of the base, &

h – height of the prism.h – height of the prism.

ExEx: find the volume of the prism.: find the volume of the prism.

V = BhV = Bh

B = ½ bh = ½ (2.5)(6) = 7.5B = ½ bh = ½ (2.5)(6) = 7.5

V = (7.5)(3)V = (7.5)(3)

V = 22.5 mmV = 22.5 mm33

6 m

m6

mm

2.5 mm2.5 mm 3 mm

3 mm

Thm 12.8Thm 12.8 – Volume of a Cylinder – Volume of a CylinderV = BhV = Bh

OrOr

V = V = rr22hh

ExEx: Find the vol. of the cylinder.: Find the vol. of the cylinder.

V = V = rr22hh

V = V = (4(422)(13))(13)

V = V = (16)(13)(16)(13)

V = 208V = 208 ft ft33

or 653.45 ftor 653.45 ft33

13 ft13 ft

4 ft

4 ft

ExEx: If the volume of the prism is 396 cm: If the volume of the prism is 396 cm33, , then what is the value of x?then what is the value of x?

V = BhV = Bh

396 = (11x)(x)396 = (11x)(x)

396 = 11x396 = 11x22

36 = x36 = x22

6 = x6 = x

x = 6 cmx = 6 cm

11 cm11 cm

x c

mx

cm

x cmx cm

ExEx: Find the : Find the volume of the volume of the

concrete block.concrete block.

(whole block) V = Bh = (13.5*4)(3) = 162 in(whole block) V = Bh = (13.5*4)(3) = 162 in22

(1 hole) V = Bh = (3*6)(3) = 54 in(1 hole) V = Bh = (3*6)(3) = 54 in33

(2 holes) V = 2(54) = 108 in(2 holes) V = 2(54) = 108 in33

Volume of concreteVolume of concrete = 162 – 108 = 54 in = 162 – 108 = 54 in33

* If concrete weighs 0.084 lbs/in* If concrete weighs 0.084 lbs/in33, how much , how much does this block weigh?does this block weigh?

0.084 * 54 = 4.536 lbs0.084 * 54 = 4.536 lbs

6 in6 in3 in3 in

13.5 in13.5 in

3 in3 in

4 in4 in

HintHint: Find the volume of : Find the volume of the whole block 1the whole block 1stst, then , then subtract out the volume of subtract out the volume of the holes!the holes!