1. 3-D Shapes Topic 14: Lesson 10
3-D Shapes Topic 14: Lesson 10 Volume of Cones Holt Geometry Texas ©2007

2. Objectives and Student Expectations
TEKS: G2B, G3B, G6B, G8D, G11D
The student will make conjectures about 3-D figures and determine the validity using a variety of approaches.
The student will construct and justify statements about geometric figures and their properties.
The student will use nets to represent and construct 3-D figures.
The student will find surface area and volume of prisms, cylinders, cones, pyramids, spheres, and composite figures.
The student will describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed.

4. Example: 1
Find the volume of a cone with radius 7 cm and height 15 cm. Give your answers both in terms of  and rounded to the nearest tenth.
Volume of a pyramid
Substitute 7 for r and 15 for h.
V = 245 cm3 ≈ cm3
Simplify.

5. Example: 2 Find the volume of a cone.
Step 1 Use the Pythagorean Theorem to find the height.
162 + h2 = 342
Pythagorean Theorem
h2 = 900
Subtract 162 from both sides.
h = 30
Take the square root of both sides.

6. Example: 2 cont. Find the volume of a cone.
Step 2 Use the radius and height to find the volume.
Volume of a cone
Substitute 16 for r and 30 for h.
V  2560 cm3  cm3
Simplify.

7. Example: 3
Find the volume of the composite figure. Round to the nearest tenth.
The volume of the upper cone is

8. Example: 3 continued
Find the volume of the composite figure. Round to the nearest tenth.
The volume of the cylinder is
Vcylinder = r2h = (21)2(35)=15,435 cm3.
The volume of the lower cone is
The volume of the figure is the sum of the volumes.
V = 5145 + 15,435 + 5,880 = 26,460  83,126.5 cm3

9. Example: 4
Find the volume of a cone with base circumference 25 in. and a height 2 in. more than twice the radius.
Step 1 Use the circumference to find the radius.
2r = 25
Substitute 25 for the circumference.
r = 12.5
Solve for r.
Step 2 Use the radius to find the height.
h = 2(12.5) + 2 = 27 in.
The height is 2 in. more than twice the radius.

10. Example: 4 continued
Find the volume of a cone with base circumference 25 in. and a height 2 in. more than twice the radius.
Step 3 Use the radius and height to find the volume.
Volume of a pyramid.
Substitute 12.5 for r and 27 for h.
V =  in3 ≈ in3
Simplify.

11. Example: 5 Find the volume of the cone. Volume of a cone
Substitute 9 for r and 8 for h.
V ≈ 216 m3 ≈ m3
Simplify.

12. Example: 6
The diameter and height of the cone are divided by 3. Describe the effect on the volume.
original dimensions:
radius and height divided by 3:
Notice that If the radius and height are divided by 3, the volume is divided by 33, or 27.

13. Example: 7
The radius and height of the cone are doubled. Describe the effect on the volume.
original dimensions:
radius and height doubled:
The volume is multiplied by 8, or by 2 cubed.

14. Example: 8
A frustrum is a part of a cone that has been cut off by a plane that is parallel to the base of the cone. Find the volume of the frustrum in the figure shown if the radius of the smaller cone is 3 cm with slant height 5 cm, and the radius of the larger cone is 9 cm with slant height 15 cm. Round to the nearest tenth.
frustrum

15. Example: 8 continued frustrum
The radius of the smaller cone is 3 cm with slant height 5 cm, and the radius of the larger cone is 9 cm with slant height 15 cm. Use the Pythagorean Theorem to find the height of each cone.
frustrum