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 PrincipleIf 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 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.

• Sphere - V=4/3r3

h

Example• Find the volume of

the right prism.

V = Bh Volume of a prism formula

A = ½ bh Area of a triangleA = ½ (3)(4) Substitute valuesA = 6 cm2 Multiply values -- base

V = (6)(2) Substitute valuesV = 12 cm3 Multiply values & solve

Example• Find the volume of

the right cylinder.

V = Bh Volume of a prism formula

A = r2 Area of a circleA = 82 Substitute valuesA = 64 in.2 Multiply values -- base

V = 64(6) Substitute valuesV = 384in.3

Multiply values & solveV = 1206.37 in.3 Simplify

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

• Find the volume of a square pyramid with base edges of 15cm & a height of 22cm.

22cm

15cm

15cm

V = (⅓)Bh

= (⅓)l•w•h

= (⅓)15•15•22

= (⅓)4950

= 1650cm3

Square

Example

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

11in V = ⅓Bh

= (⅓)r2h

= (⅓)(3)2(11)

= (⅓)99

= 33 = 103.7in3

Circle

3in

Example

Ex. 6: If the radius of the cone in Ex. 5 is 7 m, what is its height?

Ex. 5: If the volume of the cylinder is 441π m3, what is the volume of the cone?

3m3

441V

3

49147

h

3m147

hrVcylinder 23

2 hrVcone

Recall:

h 49441

hm9

Ex.4: Volume of a Composite Figure

8cm

10cm

4cm

Volume of Cone first!

Vc = ⅓Bh

= (⅓)r2h

= (⅓)(8)2(10)

= (⅓)(640)

= 213.3 = 670.2cm3

Volume of Cylinder NEXT!

Vc = Bh

= r2h

= (8)2(4)

= 256 = 804.2cm3

VT = Vc + Vc

VT = 670cm3 + 804.2cm3

VT = 1474.4cm3

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

radius.

10cm

r

V = ⅓Bh

V = ⅓(r2)h

110 = (⅓)r2(10)

110 = (⅓)r2(10)

11 = (⅓)r2

33 = r2

r = √(33) = 5.7cm

Example

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

V = 36 ft3 or 113.1 ft3

343

3V

Example

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

34

3V r

342304

3r

31728 r

12 r