Shapes and Volume

Students explore the volume of three-dimensional shapes, connecting it to the operations of multiplication and addition, as well as classify two-dimensional shapes hierarchically.

Unit Summary

In Unit 3, students will explore volume of three-dimensional shapes (5.MD.3—5), connecting it to the operations of multiplication and addition (5.NBT.5, 4.NBT.4). They also use their understanding that they gradually built in prior grade levels to classify shapes in a hierarchy, seeing that attributes of shapes in one category belong to shapes in all subcategories of that category (5.G.3—4). 

In prior grade levels, students explored the idea of volume informally, comparing the capacity of various containers as being able to “hold more” or “hold less” (K.MD.2). Students have also explored one-dimensional and two-dimensional measurements of figures, developing a deep understanding of length in Grade 2 and of area in Grade 3. In their exploration of area in Grade 3, students come to understand area as an attribute of plane figures (3.MD.5) and measure it by counting unit squares (3.MD.6), and they connect area to the operations of multiplication and addition (3.MD.7). 

Students have also explored two-dimensional shapes and their attributes extensively in previous grades. “From Kindergarten on, students experience all of the properties of shapes that they will study in Grades K–7, recognizing and working with these properties in increasingly sophisticated ways” (Geometry Progression, p. 3). In Kindergarten through Grade 2, students focused on building understanding of shapes and their properties. In Grade 3, students started to conceptualize shape categories, in particular quadrilaterals. In Grade 4, work with angle measure (4.MD.5—7) lent itself to classifying figures based on the presence or absence of parallel and perpendicular sides. 

Thus, this unit builds off of students’ well-established understanding of geometry and geometric measurement. Similar to students’ work with area, students develop an understanding of volume as an attribute of solid figures (5.MD.3) and measure it by counting unit cubes (5.MD.4). Students then connect volume to the operation of multiplication of length, width, and height or of the area of the base and the height and to the operation of addition to find composite area (5.MD.5). Throughout Topic A, students have an opportunity to use appropriate tools strategically (MP.5) and make use of structure of three-dimensional figures (MP.7) to draw conclusions about how to find the volume of a figure. 

Students then move on to classifying shapes into categories and see that attributes belonging to shapes in one category are shared by all subcategories of that category (5.G.3). This allows students to create a hierarchy of shapes over the course of many days (5.G.4). Throughout this topic, students use appropriate tools strategically (MP.5) to verify various attributes of shapes including their angle measure and presence of parallel or perpendicular lines, as well as attend to precision in their use of language when referring to geometric figures (MP.6). They also look for and make use of structure to construct a hierarchy based on properties (MP.7).

In Grade 6, students will explore concepts of length, area, and volume with more complex figures, such as finding the area of right triangles or finding the volume of right rectangular prisms with non-whole-number measurements (6.G.1, 6.G.2). Students will even rely on their understanding of shapes and their attributes to prove various geometric theorems in high school (GEO.G-CO.9—11). Thus, this unit provides a nice foundation for connections in many grades to come.

Pacing: 16 instructional days (14 lessons, 1 flex day, 1 assessment day)

For guidance on adjusting the pacing for the 2020-2021 school year due to school closures, see our 5th Grade Scope and Sequence Recommended Adjustments.

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This assessment accompanies Unit 3 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep


Intellectual Prep for All Units

  • Read and annotate “Unit Summary” and “Essential Understandings” portion of the unit plan. 
  • Do all the Target Tasks and annotate them with the “Unit Summary” and “Essential Understandings” in mind. 
  • Take the unit assessment.

Essential Understandings


  • Volume refers to the amount of space a three-dimensional figure takes up. Two-dimensional figures have no volume. 
  • You can find the volume of a rectangular prism by counting individual cubic units; counting the number of cubic units in a “layer” and multiplying by the number of layers; or multiplying the length, width, and height of the figure. The latter two strategies correspond to the formulas $${v = b \times h}$$ and $${v = l \times w \times h}$$
  • You can calculate the volume of a rectangular prism by multiplying edge lengths in any order because of the associative property. 
  • Two-dimensional figures are classified by their properties into categories but can fit into more than one category at the same time. 



cubic units

regular polygon




rectangular prism


unit cube

Related Teacher Tools:

Unit Materials, Representations and Tools


Lesson Map

Topic A: Volume of Three-Dimensional Figures

Topic B: Classification of Two-Dimensional Shapes

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards


  • 5.G.B.3 — Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

  • 5.G.B.4 — Classify two-dimensional figures in a hierarchy based on properties.

Measurement and Data
  • 5.MD.C.3 — Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

  • 5.MD.C.3.A — A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume.

  • 5.MD.C.3.B — A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.

  • 5.MD.C.4 — Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

  • 5.MD.C.5 — Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

  • 5.MD.C.5.A — Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

  • 5.MD.C.5.B — Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems.

  • 5.MD.C.5.C — Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

Foundational Standards


  • 2.G.A.1

  • 3.G.A.1

  • 4.G.A.2

Measurement and Data
  • 3.MD.C.5

  • 3.MD.C.6

  • 3.MD.C.7

  • 4.MD.A.3

Number and Operations in Base Ten
  • 5.NBT.B.5

Operations and Algebraic Thinking
  • 3.OA.B.5

Future Standards


  • 6.G.A.1

  • 6.G.A.2

Standards for Mathematical Practice

  • CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

  • CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

  • CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

  • CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

  • CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

  • CCSS.MATH.PRACTICE.MP6 — Attend to precision.

  • CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

  • CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.