ExerciseIf a pyramid and a cone have bases with the same area and altitudes that are equal, are their surface areas equal?

no

If a pyramid and a cone have bases with the same area and altitudes that are equal, are their volumes equal?

yes

Exercise

In this text, what is the difference between h and H?

h = length of the altitude of a plane figure and H = length of the altitude of a solid figure.

Exercise

What would the calculation of bhH give?

the volume of a triangular prism

12

Exercise

h

wl

hwl

Formula: Volume of a Pyramid or a ConeV = BH The volume of a pyramid or cone (V) is equal to one- third the area of the base (B) times

the height (H).

13

Find the volume of the square pyramid.

= 256 cm3

V = BH13

= (82)(12)13

8 cm 8 cm

12 cm

Example 1

Find the volume of the cone.

≈ 100.5 cm3

V = BH13

= p(42)(6)13

= 32p= 32(3.14)

6 cm

4 cm

Example 2

What is the volume of a pyramid if its height is 10 units and its base is 8 units by 12 units?

320 units3

Example

What would happen to the volume of the pyramid in the previous question if its length were doubled?

The volume would be doubled.

Example

What would happen to the volume if any single dimension were doubled?

Example

The volume would be doubled.

What would happen if all the dimensions were doubled?

The volume would be multiplied by a factor of 23 = 8.

Example

What is the volume of a square pyramid if each side of its base is 6 units and its height is 5 units?

60 units3

Example

What would happen to the volume of the pyramid in the previous question if the sides of the square base were doubled?

The volume would be multiplied by a factor of 22 = 4.

Example

Formula: Volume of a SphereV = pr3 The volume of a

sphere (V) is equal to the product of , p, and the radius cubed (r).

43

43

Find the volume of a sphere with a diameter of 15 ft. to the nearest hundredth. Find the number of gallons it will hold. (1 ft.3 = 7.48 gal.)

r = = 7.5 ft.152

Example 3

V = pr343

= p(7.53)43

= p(421.875)43

= p1,687.53

≈ 1,766.25 ft.3

Example 3

≈ 13,212 gal.7.48(1,766.25)

Example 3

Find the radius of a sphere with a volume of 288p m3.

V = pr343

pr3 = 288p43

pr3 = (288p)43

34 ( ) 3

4

Example 4

pr3 = 216p

r3 = 216r = 6 m

Example 4

What is the volume of a sphere with a radius of 6 units?

Example

904.32 units3

A city needs a 10,000 m3 water tower for its increasing population. What should the radius be if the water tower is in the form of a sphere?

Example

13.37 m

A grain storage bin is a steel cylinder with a conical top. One company markets a bin that is 18’ in diameter, 16’ high at the eaves, and 21’ high at the peak.

Exercise

What is the maximum number of bushels of wheat (rounded to the nearest bushel) that can be stored in the bin? There are 0.8 bushels in one cubic foot.

Exercise

V = pr2H + pr2H13

= 1,296p + 135p = 1,431p ft.3

= p(92)(16) + p(92)(5) 13

= 1,431p ft.3 0.8 bu.1 ft.3( )

≈ 3,595 bu.

Exercise