Students apply algebraic and proportional reasoning skills to investigate angle relationships, circle measurements, uniqueness of triangles, and solid figure application problems.

Unit Summary

In Unit 6, seventh-grade students cover a range of topics from angle relationships to circles and polygons to solid figures. The seventh-grade Geometry standards are categorized as additional standards, however, there are several opportunities throughout the unit where students are engaged in the major work of the grade. In the beginning of the unit, students use and solve equations to represent relationships between angles and find missing angle measures. Investigating circles, students discover the proportional relationship between the circumference of a circle and its diameter, and understand π as the ratio of these two quantities. Students will also use their expressions skills to write numerical expressions that can be used to find surface area and volume of three-dimensional figures. 

Throughout the unit, students encounter several vocabulary words, such as complementary angles, vertical angles, radius, and circumference. Many of these words enable students to be more precise in their communications with each other (MP.6). Students will also encounter complex diagrams of angles and 3-D figures where they will need to understand what information they can glean from the diagram and plan a solution pathway before jumping in (MP.1). Students should have access to several tools they may opt to use throughout the unit, including rulers, protractors, compasses, and reference sheets (MP.5).

The foundational skills for the standards in this unit stem from fourth through sixth grades. In fourth grade, students studied the concepts of angle measurement and understood angle measure to be additive. In fifth grade, students developed an understanding of three-dimensional volume, which they further built on in sixth grade. Sixth-grade students also began to distinguish between the three-dimensional space an object takes up and the surface area that covers it.

In eighth grade, students will zoom in on right triangles and apply the Pythagorean theorem to determine side lengths in right triangles. They will also continue solving real-life applications of surface area and volume, with the addition of cones, spheres, and cylinders.

Pacing: 23 instructional days (21 lessons, 1 flex day, 1 assessment day)

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Summer Packets

Packet of problems designed to help prepare students for the start of next school year.


This assessment accompanies Unit 6 and should be given on the suggested assessment day or after completing the unit.

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Expanded Assessment Package

Additional assessment tools to help you monitor student learning and identify any skill or knowledge gaps.

Learn how to use these tools with our Guide to Assessments


Unit Prep

Essential Understandings


  • When two lines intersect, a pair of congruent vertical angles are created. This angle relationship, along with complementary and supplementary angle relationships, can be used to determine missing angle measures in diagrams. 
  • A circle is a closed shape that is defined by the set of points that are the same distance from the center of the circle. The distance from the center to any point on the circle is called the radius, and the distance across the circle through the center is called the diameter. The measurement around a circle is called the circumference and is proportional to the diameter of the circle with a constant of proportionality equivalent to $${\pi}$$. The area of a circle can be found using the formula $$A={\pi} r^2$$.
  • In any triangle, the sum of any two side lengths must be longer than the measure of the third side. Given different conditions about the side and angle measures of a triangle, one unique triangle may be formed, more than one triangle may be formed, or no triangle may be formed.  



vertical angles

adjacent angles




complementary angles

Triangle Inequality Theorem


surface area


supplementary angles

Related Teacher Tools:

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Vocabulary Package

Additional vocabulary tools that help reinforce and support student vocabulary development.

Unit Materials, Representations and Tools


  • Rulers
  • Compasses
  • Protractors
  • Graph paper
  • String
  • AngLegs or prepared wooden skewers
  • circles handout

Intellectual Prep


Internalization of Standards via the Unit Assessment

  • Take unit assessment. Annotate for: 
    • Standards that each question aligns to
    • Strategies and representations used in daily lessons
    • Relationship to Essential Understandings of unit 
    • Lesson(s) that assessment points to

Internalization of Trajectory of Unit

  • Read and annotate "Unit Summary."
  • Notice the progression of concepts through the unit using the Lesson overview.
  • Do all target tasks. Annotate the target tasks for: 
    • Essential understandings
    • Connection to assessment questions 
  • Identify key opportunities to engage students in academic discourse. Read through our Guide to Academic Discourse and refer back to it throughout the unit.

Unit-Specific Intellectual Prep

Lesson Map

Topic A: Angle Relationships

Topic B: Circles

Topic C: Building Polygons and Triangles

Topic D: Solid Figures

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards


  • 7.G.A.2 — Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

  • 7.G.A.3 — Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

  • 7.G.B.4 — Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

  • 7.G.B.5 — Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

  • 7.G.B.6 — Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Foundational Standards


  • 5.G.B.3

  • 6.G.A.1

  • 6.G.A.2

  • 6.G.A.4

  • 7.G.A.1

  • 7.G.B.6

Measurement and Data
  • 4.MD.A.3

  • 4.MD.C.5

  • 4.MD.C.6

  • 4.MD.C.7

  • 5.MD.C.3

  • 5.MD.C.5

Ratios and Proportional Relationships
  • 7.RP.A.2

Future Standards


  • G.CO.B.7

  • G.CO.B.8

  • 8.G.A.5

  • 8.G.B.6

  • 8.G.B.7

  • 8.G.C.9

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.