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.
![Photograph of a zig zag crystal vase](https://staging-cms-im.s3.amazonaws.com/gyBUid7Teg9PVhD5C9eTa1HT?response-content-disposition=inline%3B%20filename%3D%22G5.C9.PP.zigzagvase.jpg%22%3B%20filename%2A%3DUTF-8%27%27G5.C9.PP.zigzagvase.jpg&response-content-type=image%2Fjpeg&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T083152Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=1fbbd4fed79ef50607ccffdbdd31036fde0798f7202af5510e5a45dd471daa22)
- 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
\(\frac13\)