# MA.912.GR.4.5

Solve mathematical and real-world problems involving the volume of three-dimensional figures limited to cylinders, pyramids, prisms, cones and spheres.

### Examples

Example: A cylindrical swimming pool is filled with water and has a diameter of 10 feet and height of 4 feet. If water weighs 62.4 pounds per cubic foot, what is the total weight of the water in a full tank to the nearest pound?

### Clarifications

Clarification 1: Instruction includes concepts of density based on volume.

Clarification 2: Instruction includes using Cavalieri’s Principle to give informal arguments about the formulas for the volumes of right and non-right cylinders, pyramids, prisms and cones.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

• Cone
• Cylinder
• Prism
• Pyramid
• Sphere

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In middle grades, students determined the volume of right rectangular prisms and right cylinders. In Geometry, students explore for the first time the volume of pyramids, cones, and spheres. In later courses, student learn more advanced methods for calculating volume.
• Instruction includes reviewing units and conversions within and across different measurement systems (as this was done in middle grades).
• Instruction includes discussing the convenience of answering with exact values (e.g., the simplest radical form or in terms of pi) or with approximations (e.g., rounding to the 22 nearest tenth or hundredth or using 3.14, $\frac{\text{22}}{\text{7}}$ or other approximations for pi). It is also important to explore the consequences of rounding partial answers on the accuracy or precision of the final answer, especially when working in real-world contexts.
• Instruction includes reviewing the definition of cylinders, pyramids, prisms, cones and spheres (as this was done in grade 5), and discussing the definitions of right and oblique polyhedrons, cubes, tetrahedrons, regular prisms and regular pyramids.
• The population or material density based on volume is calculated by the quotient of the total population or material and the volume (i.e., population density of fish in a spherical aquarium or density of salt in a bucket of water). Have students practice finding the population or material density or the total population or material amount, given the dimensions of a three-dimensional figure. That is, part of their work includes finding the volume based on the dimensions. (MTR.7.1)
• Instruction includes the connection to two-dimensional cross-sections of three-dimensional figures to explore Cavalieri’s Principle, which states that if in two solids of equal height, the cross-sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal. (MTR.5.1)
• For example, have students compare the volume of two stacks of pennies of the same height, one organized in a straight column and the other one, one penny on top of the other, but in a slanted stack. Discuss the shape of their cross-sections at the same height and what happens with their volumes.
• For example, have students discuss how this principle is applied in the calculation of volumes of non-right (oblique) three-dimensional figures.
• For example, have students discuss how this principle can be used to find the volume of a non-right cylinder given a right cylinder with the same height and same cross-sections. (MTR.4.1
• Instruction includes exploring a variety of real-world situations where finding the volume or volume density is relevant for different purposes. Problem types include components like percentages, cost and budget, constraints, comparisons, BTUs, nutrition (e.g., calories per cup), moisture content (e.g., ounces of water in a gallon of honey) or others.
• Problem types include finding missing dimensions given the volume of a three-dimensional figure or finding the volume of composite figures.

### Common Misconceptions or Errors

• Students may not be careful with units of measurement involving volume, particularly when converting from one unit to another.
• For example, since there are approximately 25.4 millimeters in an inch, a student may incorrectly conclude that there are 25.4 cubic millimeters in a cubic inch.

• When filling cylindrical silos, the top cone is not filled. However, if the silo has a bottom cone, it is filled. Three different silos are shown in the image below.
• Part A. In silo 3, the top and bottom cones are congruent. How much more grain could silo 3 hold than silo 1?
• Part B. The diameter of silo 1 is 80% the diameter of silo 2. Is the capacity of silo 1 80% the capacity of silo 2?

• The radius of a sphere is 4 units so its volume is $\frac{\text{256}}{\text{3}}$π  cubic units.
• Part A. Discuss the value of this kind of answer for its accuracy and precision.
• Part B. Discuss the effect of replacing π in the formulas with 3.14, 3.1416, $\frac{\text{22}}{\text{7}}$ and other approximations. What happens with the answer, the volume of the figure, in each case?

### Instructional Items

Instructional Item 1
• Joshua is going to create a garden border around three sides of his backyard deck using cinder blocks. He is going to plant a flower in each hole of the cinder block. The dimensions of the cinder blocks are 8 inches by 16 inches by 8 inches. Each hole needs to be completely filled with potting soil before the flowers can be planted. Potting soil is sold in 1 cubic foot bags.
•  Part A. What are the dimensions of a cinder block hole?
•  Part B. The patio is a square with a side length of 8 feet. One of the sides of the square patio is adjacent to an exterior wall of the house. If Joshua puts blocks around the other three sides of the patio, how many bags will Joshua need to purchase to fill the blocks?
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912065: Access Geometry (Specifically in versions: 2015 - 2022, 2022 and beyond (current))
1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.GR.4.AP.5: Solve mathematical or real-world problems involving the volume of three-dimensional figures limited to cylinders, pyramids, prisms, or cones.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Volume of a Cylinder:

Students are asked to derive and explain a formula for the volume of a cylinder given a prism with the same height and the same cross-sectional area at every height.

Type: Formative Assessment

Estimating Volume:

Students are asked to model a tree trunk with geometric solids and to use the model to estimate the volume of the tree trunk.

Type: Formative Assessment

Volume of a Cone:

Students are asked to derive and explain a formula for the volume of a cone given a pyramid with the same height and the same cross-sectional area at every height.

Type: Formative Assessment

Mudslide:

Students are asked to create a model to estimate volume and mass.

Type: Formative Assessment

Volume of a Pyramid:

Students are guided through the process of writing an informal argument for the volume of a pyramid formula using Cavalieri’s Principle.

Type: Formative Assessment

Sugar Cone:

Students are asked to solve a problem that requires calculating the volume of a cone.

Type: Formative Assessment

Louvre Pyramid:

Students are asked to find the height of a square pyramid given the length of a base edge and its volume.

Type: Formative Assessment

Cylinder Formula:

Students are asked to write the formula for the volume of a cylinder, explain what each variable represents, and label the variables on a diagram.

Type: Formative Assessment

Cone Formula:

Students are asked to write the formula for the volume of a cone, explain what each variable represents, and label the variables on a diagram.

Type: Formative Assessment

Burning Sphere:

Students are asked to solve a problem that requires calculating the volume of a sphere.

Type: Formative Assessment

Chilling Volumes:

Students are asked to solve a problem involving the volume of a composite figure.

Type: Formative Assessment

Sphere Formula:

Students are asked to write the formula for the volume of a sphere, explain what each variable represents, and label the variables on a diagram.

Type: Formative Assessment

Pyramid Formula:

Students are asked to write the formula for the volume of a pyramid, explain what each variable represents, and label the variables on a diagram.

Type: Formative Assessment

Snow Cones:

Students are asked to solve a problem that requires calculating the volumes of a cone and a cylinder.

Type: Formative Assessment

Sports Drinks:

Students are asked to solve a problem that requires calculating the volume of a large cylindrical sports drink container and comparing it to the combined volumes of 24 individual containers.

Type: Formative Assessment

The Great Pyramid:

Students are asked to find the height of the Great Pyramid of Giza given its volume and the length of the edge of its square base.

Type: Formative Assessment

Do Not Spill the Water!:

Students are asked to solve a problem that requires calculating the volumes of a sphere and a cylinder.

Type: Formative Assessment

## Lesson Plans

How Many Cones Does It Take?:

This lesson is a "hands-on" activity. Students will investigate and compare the volumes of cylinders and cones with matching radii and heights. Students will first discover the relationship between the volume of cones and cylinders and then transition into using a formula to determine the volume.

Type: Lesson Plan

Filled to Capacity!:

This is a lesson where students investigate, compare, dissect, and use the relationship between volume of a cone and cylinder with equal corresponding dimensions.

Type: Lesson Plan

The Relationship Between Cones and Cylinders:

Students create a cone and a cylinder with the same height and base. At the conclusion of the lesson, the students will know that the volume ratio between the cone and cylinder is 1:3.

Type: Lesson Plan

Exploring Cavalieri's Principle:

Students will explore Cavalieri's Principle using technology. Students will calculate the volume of oblique solids and determine if Cavalieri's Principle applies.

Students will also perform transformations of a base figure in a 3-dimensional coordinate system to observe the creation of right and oblique solid figures. After these observations, students will create a conjecture about calculating the volume of the oblique solids. Students will use the conjecture to determine situations in which Cavalieri's Principle applies and then calculate the volume of various oblique solids.

Type: Lesson Plan

Students will investigate the formula for the volume of a pyramid and/or cone and use those formulas to calculate the volume of other solids. The students will have hands-on discovery working with hollow Geometric Solids that they fill with dry rice, popcorn, or another material.

Type: Lesson Plan

Cape Florida Lighthouse: Lore and Calculations:

The historic Cape Florida Lighthouse, often described as a conical tower, teems with mathematical applications. This lesson focuses on the change in volume and lateral surface area throughout its storied existence.

Type: Lesson Plan

Propensity for Density:

Students apply concepts of density to situations that involve area (2-D) and volume (3-D).

Type: Lesson Plan

Area to Volume Exploration:

In this student-centered lesson, the formulas for the volume of a cylinder, cone, and a sphere are examined and practiced. The relationship between the volume of a cone and a cylinder with the same radius and height is explored. Students will also solve real-world problems involving these three-dimensional figures.

Type: Lesson Plan

Pack It Up:

Students use geometry formulas to solve a fruit growing company's dilemma of packing fruit into crates of varying dimensions. Students calculate the volume of the crates and the volume of the given fruit when given certain numerical facts about the fruit and the crates.

Type: Lesson Plan

This lesson explores the formulas for calculating the volume of cylinders, cones, pyramids, and spheres.

Type: Lesson Plan

The Cost of Keeping Cool:

Students will find the volumes of objects. After decomposing a model of a house into basic objects students will determine the cost of running the air conditioning.

Type: Lesson Plan

In this activity, students will utilize measurement data provided in a chart to calculate areas, volumes, and densities of cookies. They will then analyze their data and determine how these values can be used to market a fictitious brand of chocolate chip cookie. Finally, they will integrate cost and taste into their analyses and generate a marketing campaign for a cookie brand of their choosing based upon a set sample data which has been provided to them.

Type: Lesson Plan

Victorious with Volume:

In this lesson, the students will explore and use the relationship of volume for cylinders and cones that have equal heights and radii.

Type: Lesson Plan

M&M Soup:

This is the informative part of a two-lesson sequence. Students explore how to find the volume of a cylinder by making connections with circles and various real-world items.

Type: Lesson Plan

## Original Student Tutorials

Volume of Spherical Bubble Tea:

Learn how to calculate the volume of spheres while learning how they make Bubble Tea in this interactive tutorial.

Type: Original Student Tutorial

I Scream! You Scream! We All Scream for... Volume!:

Learn to calculate the volume of a cone as you solve real-world problems in this ice cream-themed, interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Experts

Velocity of the Aucilla River:

Harley Means discusses the mathematical methods hydrologists use to calculate the velocity of rivers.

Type: Perspectives Video: Expert

Carbon Foam and Geometry:

Carbon can take many forms, including foam! Learn more about how geometry and the Monte Carlo Method is important in understanding it.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Unit Rate and Florida Cave Formation:

How long does it take to form speleothems in the caves at Florida Caverns State Parks?

Type: Perspectives Video: Professional/Enthusiast

Volume and Surface Area of Pizza Dough:

Michael McKinnon of Gaines Street Pies explains how when making pizza the volume is conserved but the surface area changes.

Type: Perspectives Video: Professional/Enthusiast

Mathematically Optimizing 3D Printing:

Did you know that altering computer code can increase 3D printing efficiency? Check it out!

Type: Perspectives Video: Professional/Enthusiast

Design Process for a Science Museum Exhibit:

Go behind the scenes and learn about science museum exhibits, design constraints, and engineering workflow! Produced with funding from the Florida Division of Cultural Affairs.

Type: Perspectives Video: Professional/Enthusiast

Estimating Oil Seep Production by Bubble Volume:

You'll need to bring your computer skills and math knowledge to estimate oil volume and rate as it seeps from the ocean floor. Dive in!

Type: Perspectives Video: Professional/Enthusiast

KROS Pacific Ocean Kayak Journey: Food Storage Mass and Volume:

What do you do if you don't have room for all your gear on a solo ocean trek? You're gonna need a bigger boat...or pack smarter with math.

Related Resources:
KROS Pacific Ocean Kayak Journey: GPS Data Set[.XLSX]
KROS Pacific Ocean Kayak Journey: Path Visualization for Google Earth[.KML]

Type: Perspectives Video: Professional/Enthusiast

NASA Space Flight Hardware Geometry:

If you want to take things to space, you have to have a place to put them. Just make sure they fit before you send them up.

Type: Perspectives Video: Professional/Enthusiast

## Tutorial

Find the Volume of a Triangular Prism and Cube:

This video will show to find the volume of a triangular prism, and a cube by applying the formula for volume.

Type: Tutorial

## Unit/Lesson Sequence

Three Dimensional Shapes:

In this interactive, self-guided unit on 3-dimensional shape, students (and teachers) explore 3-dimensional shapes, determine surface area and volume, derive Euler's formula, and investigate Platonic solids. Interactive quizzes and animations are included throughout, including a 15 question quiz for student completion.

Type: Unit/Lesson Sequence

## STEM Lessons - Model Eliciting Activity

Pack It Up:

Students use geometry formulas to solve a fruit growing company's dilemma of packing fruit into crates of varying dimensions. Students calculate the volume of the crates and the volume of the given fruit when given certain numerical facts about the fruit and the crates.

In this activity, students will utilize measurement data provided in a chart to calculate areas, volumes, and densities of cookies. They will then analyze their data and determine how these values can be used to market a fictitious brand of chocolate chip cookie. Finally, they will integrate cost and taste into their analyses and generate a marketing campaign for a cookie brand of their choosing based upon a set sample data which has been provided to them.

## MFAS Formative Assessments

Burning Sphere:

Students are asked to solve a problem that requires calculating the volume of a sphere.

Chilling Volumes:

Students are asked to solve a problem involving the volume of a composite figure.

Cone Formula:

Students are asked to write the formula for the volume of a cone, explain what each variable represents, and label the variables on a diagram.

Cylinder Formula:

Students are asked to write the formula for the volume of a cylinder, explain what each variable represents, and label the variables on a diagram.

Do Not Spill the Water!:

Students are asked to solve a problem that requires calculating the volumes of a sphere and a cylinder.

Estimating Volume:

Students are asked to model a tree trunk with geometric solids and to use the model to estimate the volume of the tree trunk.

Louvre Pyramid:

Students are asked to find the height of a square pyramid given the length of a base edge and its volume.

Mudslide:

Students are asked to create a model to estimate volume and mass.

Pyramid Formula:

Students are asked to write the formula for the volume of a pyramid, explain what each variable represents, and label the variables on a diagram.

Snow Cones:

Students are asked to solve a problem that requires calculating the volumes of a cone and a cylinder.

Sphere Formula:

Students are asked to write the formula for the volume of a sphere, explain what each variable represents, and label the variables on a diagram.

Sports Drinks:

Students are asked to solve a problem that requires calculating the volume of a large cylindrical sports drink container and comparing it to the combined volumes of 24 individual containers.

Sugar Cone:

Students are asked to solve a problem that requires calculating the volume of a cone.

The Great Pyramid:

Students are asked to find the height of the Great Pyramid of Giza given its volume and the length of the edge of its square base.

Volume of a Cone:

Students are asked to derive and explain a formula for the volume of a cone given a pyramid with the same height and the same cross-sectional area at every height.

Volume of a Cylinder:

Students are asked to derive and explain a formula for the volume of a cylinder given a prism with the same height and the same cross-sectional area at every height.

Volume of a Pyramid:

Students are guided through the process of writing an informal argument for the volume of a pyramid formula using Cavalieri’s Principle.

## Original Student Tutorials Mathematics - Grades 6-8

Volume of Spherical Bubble Tea:

Learn how to calculate the volume of spheres while learning how they make Bubble Tea in this interactive tutorial.

## Original Student Tutorials Mathematics - Grades 9-12

I Scream! You Scream! We All Scream for... Volume!:

Learn to calculate the volume of a cone as you solve real-world problems in this ice cream-themed, interactive tutorial.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorials

Volume of Spherical Bubble Tea:

Learn how to calculate the volume of spheres while learning how they make Bubble Tea in this interactive tutorial.

Type: Original Student Tutorial

I Scream! You Scream! We All Scream for... Volume!:

Learn to calculate the volume of a cone as you solve real-world problems in this ice cream-themed, interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Professional/Enthusiast

Estimating Oil Seep Production by Bubble Volume:

You'll need to bring your computer skills and math knowledge to estimate oil volume and rate as it seeps from the ocean floor. Dive in!

Type: Perspectives Video: Professional/Enthusiast

## Tutorial

Find the Volume of a Triangular Prism and Cube:

This video will show to find the volume of a triangular prism, and a cube by applying the formula for volume.

Type: Tutorial

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

## Perspectives Video: Professional/Enthusiast

Estimating Oil Seep Production by Bubble Volume:

You'll need to bring your computer skills and math knowledge to estimate oil volume and rate as it seeps from the ocean floor. Dive in!

Type: Perspectives Video: Professional/Enthusiast