CPALMS Logo Generated on 9/18/2025 at 12:10 AM
The webpage this document was printed/exported from can be found at the following URL:
https://www.cpalms.org/PreviewIdea/Preview/3553
Standard 4 : Use geometric measurement and dimensions to solve problems.
Cluster Standards

This cluster includes the following benchmarks.

Visit the specific benchmark webpage to find related instructional resources.

  • MA.912.GR.4.1 : Identify the shapes of two-dimensional cross-sections of three-dimensional figures.
  • MA.912.GR.4.2 : Identify three-dimensional objects generated by rotations of two-dimensional figures.
  • MA.912.GR.4.3 : Extend previous understanding of scale drawings and scale factors to determine how dilations affect the area of two-dimensional figures and the surface area or volume of three-dimensional figures.
  • MA.912.GR.4.4 : Solve mathematical and real-world problems involving the area of two-dimensional figures.
  • 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.
  • MA.912.GR.4.6 : Solve mathematical and real-world problems involving the surface area of three-dimensional figures limited to cylinders, pyramids, prisms, cones and spheres.
Cluster Information
Number:
MA.912.GR.4
Title:
Use geometric measurement and dimensions to solve problems.
Type:
Standard
Subject:
Mathematics (B.E.S.T.)
Grade:
912
Strand
Geometric Reasoning
Cluster Access Points

This cluster includes the following Access Points.

  • MA.912.GR.4.AP.1 : Identify the shape of a two-dimensional cross section of a three-dimensional figure.
  • MA.912.GR.4.AP.2 : Identify a three-dimensional object generated by the rotation of a two-dimensional figure.
  • MA.912.GR.4.AP.3 : Select the effect of a dilation on the area of two-dimensional figures and/or surface area or volume of three-dimensional figures.
  • MA.912.GR.4.AP.4 : Solve mathematical and/or real-world problems involving the area of triangles, squares, circles or rectangles.
  • 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.
  • MA.912.GR.4.AP.6 : Solve mathematical or real-world problems involving the surface area of three-dimensional figures limited to cylinders, pyramids, prisms, and cones.
Cluster Resources

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

Original Student Tutorials
Formative Assessments
  • Inside the Box: Students are asked to identify and draw cross sections of a rectangular prism and to describe their dimensions.

  • 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.

  • Area and Circumference – 1: This task is the first in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are shown a regular n-gon inscribed in a circle. They are asked to use the formula for the area of the n-gon to derive an equation that describes the relationship between the area and circumference of the circle.

  • Softball Complex: Students are asked to solve a design problem in which a softball complex is to be located on a given tract of land subject to a set of specifications.

  • Slice of a Cone: Students are asked to sketch, describe, and compare three horizontal cross sections of a cone.

  • Slice It: Students are asked to identify and describe two-dimensional cross sections of three-dimensional solids.

  • How Many Trees?: Students are asked to determine an estimate of the density of trees and the total number of trees in a forest.

  • 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.

  • Population of Utah: Students are asked to determine the population of the state of Utah given the state’s population density and a diagram of the state’s perimeter with boundary distances labeled in miles.

  • 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.

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

  • Estimating Area: Students are asked to select appropriate geometric shapes to model a lake and then use the model to estimate the surface area of the lake.

  • Area and Circumference - 3: This task is the third in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are given the definition of pi as the area of the unit circle, A(1), and are asked to use this representation of pi along with the results from the two previous tasks to generate formulas for the area and circumference of a circle.

  • Area and Circumference - 2: This task is the second in a series of three tasks that assesses the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students show that the area of the circle of radius r, A(r), can be found in terms of the area of the unit circle, A(1) [i.e., A(r) = r2 · A(1)].

  • 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.

  • Sugar Cone: Students are asked to solve a problem that requires calculating the volume of a cone.

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

  • 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.

  • 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.

  • Burning Sphere: Students are asked to solve a problem that requires calculating the volume of a sphere.

  • Windy Pyramid: Students are asked to use a net to find the surface area of a triangular pyramid.

  • Skateboard Ramp: Students are asked to draw a net of a three-dimensional figure.

  • Square Pyramid Slices: Students are asked to sketch and describe the two-dimensional figures that result from slicing a square pyramid.

  • Rectangular Prism Slices: Students are asked to sketch and describe two-dimensional figures that result from slicing a rectangular prism.

  • Prismatic Surface Area: Students are asked to determine the surface area of a right triangular prism and explain the procedure.

  • Chilling Volumes: Students are asked to solve a problem involving the volume of a composite figure.

  • 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.

  • 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.

  • Cylinder Slices: Students are asked to sketch and describe the two-dimensional figures that result from slicing a cylinder.

  • Cone Slices: Students are asked to sketch and describe the two-dimensional figures that result from slicing a cone.

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

  • 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.

  • 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.

  • Do Not Spill the Water!: Students are asked to solve a problem that requires calculating the volumes of a sphere and a cylinder.

  • 2D Rotations of Triangles: Students are given the coordinates of the vertices of a right triangle and asked to describe the solid formed by rotating the triangle about a given axis.

  • Working Backwards – 2D Rotations: Students are given a solid and asked to determine the two-dimensional shape that will create the solid when rotated about the y-axis.

  • 2D Rotations of Rectangles: Students are given the coordinates of the vertices of a rectangle and asked to describe the solid formed by rotating the rectangle about a given axis.

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.

  • Three Dimensions Unfolded: Students will use nets of prisms to find the surface area of composite three-dimensional (3-D) figures. Students will learn to identify the faces of 3-D figures that are needed to find the surface areas.

  • 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.

  • 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.

  • My Geometry Classroom: Students will learn how to find the area and perimeter of multiple polygons in the coordinate plane using the composition and decomposition methods, applying the Distance Formula and Pythagorean Theorem. Students will complete a Geometry Classroom Floor Plan group activity. Students will do a short presentation to discuss their results which leads to the realization that polygons with the same perimeter can have different areas. Students will also complete an independent practice and submit an exit ticket at the end of the lesson.

  • 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.

  • Observing the Centroid: Students will construct the medians of a triangle then investigate the intersections of the medians.

  • The Centroid: Students will construct the centroid of a triangle using graph paper or GeoGebra in order to develop conjectures. Then students will prove that the medians of a triangle actually intersect using the areas of triangles.

  • Plane Slice: Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. Students will use modeling clay to explore the cross sections that result from slicing a 3-dimensional figure.

  • Find your Formula!: 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.

  • NASA Space Shuttle Mission Patches: Students apply geometric measures and methods, art knowledge, contextual information, and utilize clear and coherent writing to analyze NASA space shuttle mission patches from both a mathematical design and visual arts perspective.

    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.

  • 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.

  • Interchangeable Wristwatch Band: Students use measures and properties of rectangular prisms and cylinders to model and rank 3D printable designs of interchangeable wristwatch bands that satisfy physical constraints.

    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.

  • Yogurt Land Container: The student will assist Yogurt Land on choosing a new size container to offer their customers. The choice of containers are different three dimensional figures. Students will revisit the concepts of volume, surface area, and profit in order to make a decision.

    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.

  • Propensity for Density: Students apply concepts of density to situations that involve area (2-D) and volume (3-D).

  • Wrapping Up Geometry (Surface Area of Triangular Prisms) : This lesson is designed to take students from recognizing nets of triangular prisms and finding areas of their individual faces, to finding the surface area of triangular prisms. 

  • 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.

  • 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.

  • Volumes about Volume: This lesson explores the formulas for calculating the volume of cylinders, cones, pyramids, and spheres.

  • 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.

  • The Grass is Always Greener: The lesson introduces area of sectors of circles then uses the areas of circles and sectors to approximate area of 2-D figures. The lesson culminates in using the area of circles and sectors of circles as spray patterns in the design of a sprinkler system between a house and the perimeter of the yard (2-D figure).

  • Poly Wants a Bridge!: "Poly Wants a Bridge" is a model-eliciting activity that allows students to assist the city of Polygon City with selecting the most appropriate bridge to build. Teams of students are required to analyze properties of bridges, such as physical composition and span length in order to solve the problem.

    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.

  • 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.

  • Can You Cut It? Slicing Three-Dimensional Figures: In this lesson, students will sketch, model, and describe cross-sections formed by a plane passing through a three-dimensional figure. Students will create a cube, right rectangular prism, and right rectangular pyramid using modeling clay dough, and then slice the model using parallel, perpendicular, and intersecting lines. Students will describe the two-dimensional figure resulting from slicing the three-dimensional model.

  • 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.

  • St. Pi Day construction with a compass & ruler: St. Pi Day construction with compass

    This activity uses a compass and straight-edge(ruler) to construct a design. The design is then used to complete a worksheet involving perimeter, circumference, area and dimensional changes which affect the scale factor ratio.

  • Turning Tires Model Eliciting Activity: The Turning Tires MEA provides students with an engineering problem in which they must work as a team to design a procedure to select the best tire material for certain situations. The main focus of the MEA is applying surface area concepts and algebra through modeling.

    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.

  • Calculating Volumes of Compound Objects: This lesson unit is intended to help you assess how well students solve problems involving measurement, and in particular, to identify and help students who have the following difficulties:

    • Computing measurements using formulas.
    • Decomposing compound shapes into simpler ones.
    • Using right triangles and their properties to solve real-world problems.
Perspectives Video: Experts
Perspectives Video: Professional/Enthusiasts
Perspectives Video: Teaching Ideas
Problem-Solving Tasks
  • Toilet Roll: The purpose of this task is to engage students in geometric modeling, and in particular to deduce algebraic relationships between variables stemming from geometric constraints.

  • Ice Cream Cone: In this task, students will provide a sketch of a paper ice cream cone wrapper, use the sketch to develop a formula for the surface area of the wrapper, and estimate the maximum number of wrappers that could be cut from a rectangular piece of paper.

  • How thick is a soda can? (Variation II): This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can. Multiple solution processes are presented.

  • How thick is a soda can? (Variation I): This problem solving task challenges students to find the surface area of a soda can, calculate how many cubic centimeters of aluminum it contains, and estimate how thick it is.

  • How many leaves on a tree? (Version 2): This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

  • How many leaves on a tree?: This is a mathematical modeling task aimed at making a reasonable estimate for something which is too large to count accurately, the number of leaves on a tree.

  • How many cells are in the human body?: This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

  • Hexagonal pattern of beehives: The goal of this task is to use geometry to study the structure of beehives.

  • Global Positioning System II: Reflective of the modernness of the technology involved, this is a challenging geometric modeling task in which students discover from scratch the geometric principles underlying the software used by GPS systems.

  • Archimedes and the King's Crown: This problem solving task uses the tale of Archimedes and the King of Syracuse's crown to determine the volume and mass of gold and silver.

  • Doctor's Appointment: The purpose of the task is to analyze a plausible real-life scenario using a geometric model. The task requires knowledge of volume formulas for cylinders and cones, some geometric reasoning involving similar triangles, and pays attention to reasonable approximations and maintaining reasonable levels of accuracy throughout.

  • Inscribing a hexagon in a circle: This problem solving task challenges students to inscribe equilateral triangles and regular hexagons on a circle with a compass and straightedge.

  • Centerpiece: The purpose of this task is to use geometric and algebraic reasoning to model a real-life scenario. In particular, students are in several places (implicitly or explicitly) to reason as to when making approximations is reasonable and when to round, when to use equalities vs. inequalities, and the choice of units to work with (e.g., mm vs. cm).

  • Use Cavalieri’s Principle to Compare Aquarium Volumes: This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere.

  • Tennis Balls in a Can: This task is inspired by the derivation of the volume formula for the sphere. If a sphere of radius 1 is enclosed in a cylinder of radius 1 and height 2, then the volume not occupied by the sphere is equal to the volume of a "double-naped cone" with vertex at the center of the sphere and bases equal to the bases of the cylinder

  • 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.

  • 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.

  • 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.

  • 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.

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.