# 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
Date Adopted or Revised: 08/20
Status: State Board Approved

## 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 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

Propensity for Density:

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

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.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

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

Which Brand of Chocolate Chip Cookie Would You Buy?:

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.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

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.

Download the CPALMS Perspectives video student note taking guide.

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?

Download the CPALMS Perspectives video student note taking guide.

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]

Download the CPALMS Perspectives video student note taking guide.

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.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

Which Brand of Chocolate Chip Cookie Would You Buy?:

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.

Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click here to learn more about MEAs and how they can transform your classroom.

## 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