Chapter 12

12.5

Volume of Pyramids and Cones

GOAL 1

GOAL 2

Finding Volumes of Cones

Using Volumes in Real-Life Problems

Find the area of the base of the cone1. Base is a circle

• Area of Circle = r2

2. B = 36 in2

Theorem 12.9 Volume of a Cone

• B is the area of the base

• H is the height

21 1

3 3V Bh r h

Find the volume of each cone

1. Area of the Base

• B = r2 = (4)2 = 16

2. Find the height

• h = 12

3. Fill in the formula

• 64)12)(16(3

1V

Find the volume of each cone1. Area of the base

• B = r2 = (5)2 = 25

2. Find the height

• From vertex perpendicular to base

• h2 + 52 = 82

• h = 39

3. Fill in the formula

• 3

3925)39)(25(

3

1 V

Find the volume of each cone

1. Area of the base

• B = r2 = (9)2 = 81

2. Find the height

• From vertex perpendicular to base

• h2 + 92 = 212

• h = 360 = 610

3. Fill in the formula

• 10162106813

1 V

10. To complete a construction job, a contractor needs 145 more cubic yards of concrete. If there remains a conical pile of concrete mix measuring 36 feet in diameter and 12 feet high, is there enough concrete still on the job site to finish the job? Explain your reasoning.

Applications

1. How much does he have? (Find the volume or the pile)

• Convert feet to yards d = 12 yds, r = 6 yds, h = 4 yds

• Find the Volume

• 32 8.15048463

1ydsV

2. Yes he has plenty

11. The limestone blocks from which an ancient pyramid was made weigh about 2 tons per cubic yard. Find the approximate weight of the pyramid having a square base of length 250 yards and a height of 150 yards

Applications

1. What is the volume (Use the formula for a pyramid)

• 32 31250001502503

1ydsV

2. Find the weight (Multiply by 2)

• 3125000(2) = 6250000 tons

Find the volume of the solid

1. Composed of two cones each with radius of 5.

2. Need to find the height of each cone.

3. Find the Volume of each cone

4. Add the volumes together

First, find the volume of the left cone

Left Cone

1. Find the height (pythagorean theorem)

• h2 + 52 = 102 h = 53

2. Find the Volume

• 32 72.2263

3125355

3

1inV

Second, find the volume of the right cone

Right Cone

1. Find the height (pythagorean theorem)

• h2 + 52 = 182 h = 299

2. Find the Volume

• 32 69.4523

299252995

3

1inV

Now find the total Volume

• Volume of left cone: V 226.72 in3

• Volume of Right Cone: V 452.69 in3

• Total Volume: V 226.72 + 452.69 = 679.41in3

Homework #67Pg 755-757 9, 17-19, 21, 25-37, 40-51