Help

MAFS.8.G.3.9

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Clarifications

Fluency Expectations or Examples of Culminating Standards

When students learn to solve problems involving volumes of cones, cylinders, and spheres — together with their previous grade 7 work in angle measure, area, surface area and volume (7.G.2.4–2.6) — they will have acquired a well-developed set of geometric measurement skills. These skills, along with proportional reasoning (7.RP) and multistep numerical problem solving (7.EE.2.3), can be combined and used in flexible ways as part of modeling during high school — not to mention after high school for college and careers.

General Information

Subject Area: Mathematics
Grade: 8
Domain-Subdomain: Geometry
Cluster: Level 2: Basic Application of Skills & Concepts
Cluster: Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. (Additional Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date Adopted or Revised: 02/14
Date of Last Rating: 02/14
Status: State Board Approved
Assessed: Yes

Test Item Specifications


  • Assessment Limits :
    Graphics of three-dimensional figures can be included. Dimensions must be given as rational numbers. Figures must not be composite
  • Calculator :

    Yes

  • Context :

    Allowable

Sample Test Items (4)

  • Test Item #: Sample Item 1
  • Question:

    A cylinder with a height of begin mathsize 12px style 6 1 half end style inches (in.) and a diameter of 5 inches is shown.

    What is the volume of the cylinder, in cubic inches? (Use 3.14 for begin mathsize 12px style straight pi end style)

  • Difficulty: N/A
  • Type: EE: Equation Editor

  • Test Item #: Sample Item 2
  • Question:

    The diameter of a sphere is 4 inches. 

    What is the volume of the sphere, in cubic inches? (Use 3.14 for ??.)

  • Difficulty: N/A
  • Type: EE: Equation Editor

  • Test Item #: Sample Item 3
  • Question:

    A cone has a height of 6.4 inches and a diameter of 6 inches.

    What is the volume, in cubic inches, of the cone? Use 3.14 for begin mathsize 12px style straight pi end style

  • Difficulty: N/A
  • Type: EE: Equation Editor

  • Test Item #: Sample Item 4
  • Question:

    A water container in the shape of a cone has a height of 5 inches and a diameter of 3.5 inches.

    Enter an equation in the first blank box and a number in the second blank box to complete the statements about the water container.

    A. The formula to calculate the container's volume, V, with the given measurements is ________.

    B. The container can hold approximately _____ cubic inches of water. Round your answer to the nearest hundredth.

     

  • Difficulty: N/A
  • Type: EE: Equation Editor

Related Courses

This benchmark is part of these courses.
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
2002100: M/J Comprehensive Science 3 (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
2002110: M/J Comprehensive Science 3, Advanced (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
2003010: M/J Physical Science (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
2003020: M/J Physical Science, Advanced (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
1204000: M/J Intensive Mathematics (MC) (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 and beyond (current))
7820017: Access M/J Comprehensive Science 3 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 and beyond (current))
2002085: M/J Comprehensive Science 2 Accelerated Advanced (Specifically in versions: 2014 - 2015, 2015 and beyond (current))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MAFS.8.G.3.AP.9a: Using a calculator, apply the formula to find the volume of three-dimensional shapes (i.e., cubes, spheres and cylinders).

Related Resources

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

Assessments

Sample 2 - Eighth Grade Math State Interim Assessment:

This is a State Interim Assessment for eighth grade.

Type: Assessment

Sample 1 - Eighth Grade Math State Interim Assessment:

This is a State Interim Assessment for eighth grade.

Type: Assessment

Formative Assessments

Sugar Cone:

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

Type: Formative Assessment

Platinum Cylinder:

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

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

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

Lesson Plans

What Floats Your Boat:

Students will solve real-world and mathematical problems involving area, density, volume and surface area of 2 – and 3-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Students will engineer solutions to the given problem using gained scientific content knowledge as application of mathematical skills

Type: Lesson Plan

Slope Intercept - Lesson #1:

This is lesson 1 of 3 in the Slope Intercept unit. This lesson introduces graphing proportional relationships. In this lesson students will perform an experiment to find and relate density of two different materials to the constant of proportionality and unit rate.

Type: Lesson Plan

Knight Shipping, Inc.:

In this design challenge students will take what they have learned about calculating the volumes and densities of cones, cylinders, and spheres, to decide which shape would make the best shipping container. Students will calculate the volumes and densities to help select the best design and then test them to move at least 3 containers of the chosen shape across the classroom. Students will fill the shapes with marshmallows to visually confirm which shape would hold more.

Type: Lesson Plan

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 radius and height. They will first discover the relationship between the volume of cones and cylinders and transition to using formulas to determine the volumes.

Type: Lesson Plan

Find your Formula!:

  • Through investigative activity, the students will determine the formula for Volume of Pyramids and/or Cones and use those formulas to calculate the volume of various solids.
  • The students will have hands on discovery working with hollow Geometric Solids and fill (dry rice, popcorn or other fill material).

Type: Lesson Plan

Silly Cylinders:

This is a short activity where students determine the density of the human body by considering each part of the body to be a cylinder. I use this activity during the second week of school, so students have already had some practice with measurement. In addition to providing students with practice in data collection and problem solving, it is a good activity that allows teachers to measure students' previous knowledge in these areas.

Type: Lesson Plan

Area to Volume Exploration:

Students will learn the formulas for the volume of a cylinder, volume of a cone, and volume of a sphere to solve real-world and mathematical problems. Students will also find volumes of hemispheres, and solids composed of spheres, hemispheres, cylinders and cones.

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

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 part one of two in a lesson sequence and is primarily formative in nature. In this lesson students will explore how to find the volume of a cylinder.

Type: Lesson Plan

Cylinder Volume Lesson Plan:

Using volume in the real world

Type: Lesson Plan

Relating Surface Area and Volume:

This lesson is enhanced through the multimedia CPALMS Perspectives Video Volume and Surface Area of Pizza Dough, which will introduce students to the relationship between surface area and volume. At the conclusion of this lesson, students will be able to relate surface area and volume, and recognize and understand that while the surface area might change, the volume can remain the same.

Type: Lesson Plan

Original Student Tutorial

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

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

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

Problem-Solving Tasks

Glasses:

In this resource, students will determine the volumes of three different shaped drinking glasses. They will need prior knowledge with volume formulas for cylinders, cones, and spheres, as well as experience with equation solving, simplifying square roots, and applying the Pythagorean theorem.

Type: Problem-Solving Task

Comparing Snow Cones:

Students will just be learning about similarity in this grade, so they may not recognize that it is needed in this context. Teachers should be prepared to give support to students who are struggling with this part of the task. To simplify the task, the teacher can just tell the students that based on the slant of the truncated conical cup, the complete cone would be 14 in tall and the part that was sliced off was 10 inches tall. (See solution for an explanation.) There is a worthwhile discussion to be had about parts (c) and (e). The percentage increase is smaller for the snow cones than it was for the juice treats. The snow cones have volume which is equal to those of the juice treats plus the volume of the dome, which is the same in both cases. Adding the same number to two numbers in a ratio will always make their ratio closer to one, which in this case means that the ratio - and thus percentage increase - would be smaller.

Type: Problem-Solving Task

Flower Vases:

The purpose of this task is to give students practice working the formulas for the volume of cylinders, cones and spheres, in an engaging context that provides and opportunity to attach meaning to the answers.

Type: Problem-Solving Task

Shipping Rolled Oats:

Students should think of different ways the cylindrical containers can be set up in a rectangular box. Through the process, students should realize that although some setups may seem different, they result in a box with the same volume. In addition, students should come to the realization (through discussion and/or questioning) that the thickness of a cardboard box is very thin and will have a negligible effect on the calculations.

Type: Problem-Solving Task

Shamu Stadium Geometry-SeaWorld Classroom Activity:

In this problem solving task, students investigate Shamu Stadium at Sea World. They will use knowledge of geometric shapes to solve problems involving area and volume and examine as well as analyze a diagram making calculations. Students will also be challenged to design an advertising poster using the measurements they mind.

Type: Problem-Solving Task

Student Center Activity

Edcite: Mathematics Grade 8:

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

Teaching Idea

Modeling: Making Matchsticks:

This lesson unit is intended to help you assess how well students are able to:

  • Interpret a situation and represent the variables mathematically.
  • Select appropriate mathematical methods.
  • Interpret and evaluate the data generated.
  • Communicate their reasoning clearly.
The context is estimating how many matchsticks (rectangular prisms) can be made from this tree (conic).

Type: Teaching Idea

Tutorials

Cylinder Volume and Surface Area:

This video demonstrates finding the volume and surface area of a cylinder.

Type: Tutorial

Volume of a Sphere:

This video shows how to calculate the volume of a sphere.

Type: Tutorial

Volume of a Cone:

This video explains the formula for volume of a cone and applies the formula to solve a problem.

Type: Tutorial

Converting Speed Units:

In this lesson, students will be viewing a Khan Academy video that will show how to convert ratios using speed units.

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.

MFAS Formative Assessments

Burning Sphere:

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

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.

Louvre Pyramid:

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

Platinum Cylinder:

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

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.

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.

Sugar Cone:

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

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.

Student Resources

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

Original Student Tutorial

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

Problem-Solving Tasks

Glasses:

In this resource, students will determine the volumes of three different shaped drinking glasses. They will need prior knowledge with volume formulas for cylinders, cones, and spheres, as well as experience with equation solving, simplifying square roots, and applying the Pythagorean theorem.

Type: Problem-Solving Task

Comparing Snow Cones:

Students will just be learning about similarity in this grade, so they may not recognize that it is needed in this context. Teachers should be prepared to give support to students who are struggling with this part of the task. To simplify the task, the teacher can just tell the students that based on the slant of the truncated conical cup, the complete cone would be 14 in tall and the part that was sliced off was 10 inches tall. (See solution for an explanation.) There is a worthwhile discussion to be had about parts (c) and (e). The percentage increase is smaller for the snow cones than it was for the juice treats. The snow cones have volume which is equal to those of the juice treats plus the volume of the dome, which is the same in both cases. Adding the same number to two numbers in a ratio will always make their ratio closer to one, which in this case means that the ratio - and thus percentage increase - would be smaller.

Type: Problem-Solving Task

Flower Vases:

The purpose of this task is to give students practice working the formulas for the volume of cylinders, cones and spheres, in an engaging context that provides and opportunity to attach meaning to the answers.

Type: Problem-Solving Task

Shipping Rolled Oats:

Students should think of different ways the cylindrical containers can be set up in a rectangular box. Through the process, students should realize that although some setups may seem different, they result in a box with the same volume. In addition, students should come to the realization (through discussion and/or questioning) that the thickness of a cardboard box is very thin and will have a negligible effect on the calculations.

Type: Problem-Solving Task

Student Center Activity

Edcite: Mathematics Grade 8:

Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.

Type: Student Center Activity

Tutorials

Cylinder Volume and Surface Area:

This video demonstrates finding the volume and surface area of a cylinder.

Type: Tutorial

Volume of a Sphere:

This video shows how to calculate the volume of a sphere.

Type: Tutorial

Volume of a Cone:

This video explains the formula for volume of a cone and applies the formula to solve a problem.

Type: Tutorial

Converting Speed Units:

In this lesson, students will be viewing a Khan Academy video that will show how to convert ratios using speed units.

Type: Tutorial

Parent Resources

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

Problem-Solving Tasks

Glasses:

In this resource, students will determine the volumes of three different shaped drinking glasses. They will need prior knowledge with volume formulas for cylinders, cones, and spheres, as well as experience with equation solving, simplifying square roots, and applying the Pythagorean theorem.

Type: Problem-Solving Task

Comparing Snow Cones:

Students will just be learning about similarity in this grade, so they may not recognize that it is needed in this context. Teachers should be prepared to give support to students who are struggling with this part of the task. To simplify the task, the teacher can just tell the students that based on the slant of the truncated conical cup, the complete cone would be 14 in tall and the part that was sliced off was 10 inches tall. (See solution for an explanation.) There is a worthwhile discussion to be had about parts (c) and (e). The percentage increase is smaller for the snow cones than it was for the juice treats. The snow cones have volume which is equal to those of the juice treats plus the volume of the dome, which is the same in both cases. Adding the same number to two numbers in a ratio will always make their ratio closer to one, which in this case means that the ratio - and thus percentage increase - would be smaller.

Type: Problem-Solving Task

Flower Vases:

The purpose of this task is to give students practice working the formulas for the volume of cylinders, cones and spheres, in an engaging context that provides and opportunity to attach meaning to the answers.

Type: Problem-Solving Task

Shipping Rolled Oats:

Students should think of different ways the cylindrical containers can be set up in a rectangular box. Through the process, students should realize that although some setups may seem different, they result in a box with the same volume. In addition, students should come to the realization (through discussion and/or questioning) that the thickness of a cardboard box is very thin and will have a negligible effect on the calculations.

Type: Problem-Solving Task