Pythagorean Theorem and Volume

Students learn about irrational numbers, approximating square roots of non-perfect square numbers, and investigate the well-known Pythagorean Theorem to solve for missing measures in right triangles.

Unit Summary

In Unit 7, eighth-grade students extend their understanding of the Number System to include irrational numbers. This new understanding supports students as they study square and cube root equations and relationships between side lengths in right triangles, both concepts that fall within the major work of the grade. Students start the unit by investigating solutions to equations like $$x^2=2$$ and realize that the solution is not an exact point on the number line. They approximate square roots of non-perfect square numbers and represent rational numbers written in decimal form as fractions. The focus of the unit shifts to right triangles, and students investigate the well-known Pythagorean Theorem. They apply their understanding of square roots to solve for missing measures in right triangles and other applications. They look closely at geometric figures to identify and create right triangles, opening up the opportunity to apply the Pythagorean Theorem to find new information (MP.7). The focus shifts once more as students learn about cube roots and apply this new concept to various volume applications involving cylinders, spheres, and cones. Throughout the unit, students must attend to precision in their work, their solutions, and their communication, being careful about specifying appropriate units of measure, using the equals sign appropriately, and representing numbers accurately (MP.6). 

Prior to this unit, students learned many skills and concepts that prepared them for this unit. Since elementary grades, students have been learning about and refining their understanding of area and volume. They have learned how to use composition and decomposition as tools to determine measurements, they’ve learned formulas and how to use them in problem-solving situations, and they’ve encountered various real-world situations. Standard 8.G.9 is a culminating standard in the Geometry progression in middle school, which will lay the foundation for much of the work they will do in high school geometry. 

In high school, students will more formally derive the distance formula and other principles, they will expand their work with right triangles to include trigonometric ratios, and they will solve more complex problems involving volume of cylinders, pyramids, cones, and spheres.

Pacing: 20 instructional days (16 lessons (17 days), 2 flex days, 1 assessment day)

For guidance on adjusting the pacing for the 2021-2022 school year, see our 8th Grade Scope and Sequence Recommended Adjustments.

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This assessment accompanies Unit 7 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep


Internalization of Standards via the Post-Unit Assessment

  • Take the Post-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 the Unit Summary.
  • Notice the progression of concepts through the unit using the Lesson Map.
  • Do all Target Tasks. Annotate the Target Tasks for: 
    • Essential Understandings
    • Connection to Post-Unit Assessment questions
  • Identify key opportunities to engage students in academic discourse. Read through our Teacher Tool on Academic Discourse and refer back to it throughout the unit.

Unit-Specific Intellectual Prep

Essential Understandings


  • The square root of $$p$$, or $$\sqrt{p}$$, represents a solution to the equation $$x{^2}=p$$. It is the measure of the side length of a square with an area of $$p$$ units$${^2}$$. The cube root of $$p$$, or $$\sqrt[3]{p}$$, represents a solution to the equation $$x{^3}=p$$. It is the measure of the side length of a cube with a volume of $$p$$ units$${^3}$$.
  • Every rational number can be expressed as a fraction $${a\over{b}}$$, where $$a$$ and $$b$$ are integers and $$b\neq0$$. As a decimal, every rational number either terminates or repeats. An irrational number has a decimal expansion that neither terminates nor repeats. For example, $${\sqrt2}$$ is an irrational number.
  • The Pythagorean Theorem describes the relationship between the side lengths of a right triangle. If a triangle is a right tringle, then $$ a{^2}+b{^2}=c{^2}$$, where $$a$$ and $$b$$ are the legs of the triangle and $$c$$ is the hypotenuse. The converse of the Pythagorean Theorem is also true. If the relationship $$ a{^2}+b{^2}=c{^2}$$ holds true for a triangle, then the triangle is a right triangle.
  • Many real-world problems can be modeled and solved using cylinders, cones, spheres, and other three-dimensional shapes.





square root

cube root

perfect square

irrational number

rational number

converse statement

pythagorean theorem

pythagorean triplet




To see all the vocabulary for this course, view our 8th Grade Vocabulary Glossary.



  • Calculators (1 per student)
  • Graph Paper (2-3 sheets per student)
  • Poster paper (1 sheet per small group)

To see more information about the materials in this unit, view the Unit Materials Overview.

Lesson Map

Topic A: Irrational Numbers and Square Roots



Define, evaluate, and estimate square roots.



Locate irrational values approximately on a number line. Compare values of irrational numbers.



Represent rational numbers as decimal expansions.



Represent decimal expansions as rational numbers in fraction form.

Topic B: Understanding and Applying the Pythagorean Theorem



Understand a proof of the Pythagorean Theorem.



Use the converse of the Pythagorean Theorem to determine if a triangle is a right triangle.



Find missing side lengths involving right triangles and apply to area and perimeter problems.



Solve real-world and mathematical problems using the Pythagorean Theorem (Part I).



Solve real-world and mathematical problems using the Pythagorean Theorem (Part II).



Find the distance between points in the coordinate plane using the Pythagorean Theorem.

Topic C: Volume and Cube Roots



Solve real-world and mathematical problems involving the volume of cylinders and cones.



Solve real-world and mathematical problems involving the volume of spheres.

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards


Expressions and Equations
  • 8.EE.A.2 — Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

  • 8.G.B.6 — Explain a proof of the Pythagorean Theorem and its converse.

  • 8.G.B.7 — Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

  • 8.G.B.8 — Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

  • 8.G.C.9 — Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

The Number System
  • 8.NS.A.1 — Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

  • 8.NS.A.2 — Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

Foundational Standards


Expressions and Equations
  • 6.EE.B.5

  • 6.G.A.1

  • 6.G.A.3

  • 7.G.B.4

  • 7.G.B.6

The Number System
  • 6.NS.C.6

  • 7.NS.A.2.D

  • 7.NS.A.3

Future Standards


Expressing Geometric Properties with Equations
  • G.GPE.B.7

Geometric Measurement and Dimension
  • G.GMD.A.1

  • G.GMD.A.2

  • G.GMD.A.3

Similarity, Right Triangles, and Trigonometry
  • G.SRT.C.6

  • G.SRT.C.7

  • G.SRT.C.8

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.