Topic: Pyramids and Cones (G-GMD.1.3)Resources: Math Nation Section 14 Videos 6 and 7 AND Textbook 12-3, 12-5Essential Question: How are pyramids and cones related to prisms?

Use the word bank below to label every part of the pyramid and cone.

Apex Height Slant Height Radius Base Lateral Surface (face)

Pyramid: Cone:

1. Consider the following cone and cylinder of equal diameter and equal height. a. What is the ratio of the cone to the cylinder based on volume?

b. How many cones does it take to fill the cylinder?

c. Consider the formula for the volume of a cylinder and derive the formula for the volume of the cone.

2. Consider the following diagram of a cone and its net. a. Determine the slant height of the cone.

b. Determine the lateral area of the cone.

c. Determine the surface area of the cone.

d. Determine the volume of the cone.

3. Consider the following diagram of a pyramid and its net. a. What specific shapes are in the net of the pyramid?

b. Determine the slant height of the pyramid.

c. Determine the lateral area of the pyramid.

d. Determine the surface area of the pyramid.

e. Determine the volume of the pyramid.

4. Copy the given measurements onto the net of the cone below and find the lateral area, surface area, and volume of the cone. Include appropriate units.

a. Lateral area ___________________

b. Surface area ___________________

c. Volume ___________________

5. Solve for the variable using the given the volume is 314.15 in3.

6. Given the square pyramid, find the lateral area, surface area, and volume. Include appropriate units.

a. Lateral area ___________________

b. Surface area ___________________

c. Volume ___________________

12 in

x

8 m

8 m

12 m

7. Given the hexagonal pyramid, find the lateral area, surface area, and volume. Include appropriate units.

a. Lateral area ___________________

b. Surface area ___________________

c. Volume ___________________

8. Given the square pyramid, find the lateral area, surface area, and volume. Include appropriate units.

Lateral Area: ________________________________

Surface Area: ________________________________

2 m

8 m

6 m

4 m

7.1 m

Volume: ________________________________

9. Given the hexagonal pyramid, find the lateral area, surface area, and volume. Include appropriate units.

Lateral Area: ________________________________

17 in

Surface Area: ________________________________

Volume: ________________________________

12 in

10. Given the cone, find the lateral area, surface area, and volume. Include appropriate units.

Lateral Area: ________________________________

Surface Area: ________________________________

13 cm

14 cm

Volume: ________________________________

11. Given the cone, find the lateral area, surface area, and volume. Include appropriate units.

Lateral Area: ________________________________

Surface Area: ________________________________

Volume: ________________________________

12. Given the cone, find the lateral area, surface area, and volume. Include appropriate units.

Lateral Area: ________________________________

Surface Area: ________________________________

15.75 mm

12.73 mm

Volume: ________________________________

13. Given the triangular pyramid, find the lateral area, surface area, and volume. Include appropriate units.

Lateral Area: ________________________________

Surface Area: ________________________________

Volume: ________________________________

14. Snow Cones: Jennifer loves snow cones and wants to get the most for her money. There are two vendors at the fair selling snow cones for the same price. If the two containers are completely filled and then leveled off across their tops, which will hold the most? If necessary, round off to the nearest cubic centimeter.

Traditional Snow Cone Snow Cone in a Cup

15. The great pyramid: The Great Pyramid of Giza is an example of a square pyramid and is the last surviving structure considered a wonder of the ancient world. The builders of the pyramid used a measure called a cubit, which represents the length of the forearm from the elbow to the tip of the middle finger. One cubit is about 20 inches in length. Find the height of the Great Pyramid (in cubits) if each base edge is 440 cubits long and the volume of the pyramid is 18,069,330 cubic cubits. Justify your response by showing and/or explaining your work.

Critical thinking: Make a conjecture about the volumes of the two pyramids, and similarly the cones.

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4 cm

13 cm

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Topic: Cavalieri’s Principle (G-GMD.1.2)Resources: Math Nation Section 12 Topic 2, 3 AND Textbook 12-4Essential Question: How does Cavalier’s Principle utilize to volume and surface area?

The volume of a solid is the number of cubic units contained in its interior. Volume is measured in cubic units, such as cubic meters, m3.

If two solids have the same ___________________ and the same ___________________ _________ at

every level, then they have the ___________________ ___________________.

1. Does this prove Cavalieri’s Principle? How can you prove it?

2. Other Examples:

3. The objective of this exercise is to provide an informal argument for the volume of a pyramid formula. One way to do this is to use Cavalieri’s Principle which states: If two solids have bases of the same area and cross-sections of the same area at any given height above the bases, then those two solids have the same volume. Note that if two pyramids have congruent bases and the same height, their cross sections at any height must be congruent. By Cavalieri’s Principle, it follows that the volumes of the pyramids are the same. Consider triangular prism ABCDEF. Suppose BD, EC , and CD are drawn as shown below.

a. What is the relationship between the area of ∆ ABD and the area of ∆ EDB? Explain.

b. Consider pyramid ABDC and pyramid EDBC. What is the relationship between their bases and heights? What is the relationship between their volumes? Explain.

c. What is the relationship between the volume of any one of these pyramids and the volume of prism ABCDEF?

Summary of Surface Area and Volume of Pyramids and Cones:

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