Building a Volume Formula for a Pyramid

ID: kasip-nojig
Favorite
Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Geometry
Grade: 9-12
PreviewAnswer Key

14 questions

Building a Volume Formula for a Pyramid

Classroom:
Due:
Student Name:
Date Submitted:
Problem 1

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

Show Work
Problem 2

A cylinder has radius 9 inches and height 15 inches. A cone has the same radius and height. Leave answers in exact form, in terms of π\pi if necessary.

2) Find the volume of the cylinder.

Show Work

3) Find the volume of the cone.

Show Work

4) What fraction of the cylinder’s volume is the cone’s volume?

Show Work
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=0.25k = 0.25.

A template for answering this question. Ask your instructor for an alternative.

5) Calculate the area of each of the 2 cross sections.

6) Suppose a new cross section was created in each solid, both at the same height, using some scale factor kk. What would the area of these cross sections be?

7) Explain your reasoning.

Problem 4

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

A template for answering this question. Ask your instructor for an alternative.
a)

Triangular pyramid

b)

Regular prism

c)

Square prism

d)

Right triangular prism

Problem 5

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

a)

Rectangular pyramid

b)

Triangular pyramid

c)

Right triangular prism

d)

Rectangular prism

Problem 6

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

A template for answering this question. Ask your instructor for an alternative.
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.

A template for answering this question. Ask your instructor for an alternative.

11) If we assume the crystal itself has no thickness, what would be the volume of the vase?

12) 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?

13) Explain or show your reasoning.

Problem 8

14) 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?

Show Work