# Lesson 13

Building a Volume Formula for a Pyramid

- Let’s create a formula for the volume of any pyramid or cone.

### Problem 1

Find the volume of a pyramid whose base is a square with side lengths of 6 units and height of 8 units.

### Problem 2

A cylinder has radius 9 inches and height 15 inches. A cone has the same radius and height.

- Find the volume of the cylinder.
- Find the volume of the cone.
- What fraction of the cylinder’s volume is the cone’s volume?

### Problem 3

Each solid in the image has height 4 units. The area of each solid’s base is 8 square units. A cross section has been created in each by dilating the base using the apex as a center with scale factor \(k=0.25\).

- Calculate the area of each of the 2 cross sections.
- Suppose a new cross section was created in each solid, both at the same height, using some scale factor \(k\). How would the areas of these 2 cross sections compare? Explain your reasoning.

### Problem 4

Select the most specific and accurate name for the solid in the image.

triangular pyramid

regular prism

square prism

right triangular prism

### Problem 5

A solid can be constructed with 4 triangles and 1 rectangle. What is the name for this solid?

rectangular pyramid

triangular pyramid

right triangular prism

rectangular prism

### Problem 6

Find the volume of the solid produced by rotating this two-dimensional shape using the axis shown.

### Problem 7

This zigzag crystal vase has a height of 20 centimeters. The cross sections parallel to the base are always rectangles that are 12 centimeters wide by 6 centimeters long.

- If we assume the crystal itself has no thickness, what would be the volume of the vase?
- The crystal is actually 1 centimeter thick on each of the sides and on the bottom. Approximately how much space is contained within the vase? Explain or show your reasoning.

### Problem 8

A trapezoid has an area of 10 square units. What scale factor would be required to dilate the trapezoid to have an area of 90 square units?

9

6

3

\(\frac13\)