Home›Lesson Plans›Mathematics›Geometry · Grade 8

The Pythagorean Theorem: a² + b² = c²

Students discover one of math’s most famous rules — in any right triangle, the squares of the two legs add up to the square of the hypotenuse — and use it to find a missing side.

Grade 8Geometry50 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
Start the Lesson
Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓State the Pythagorean theorem.
  • ✓Identify the legs and hypotenuse.
  • ✓Find the hypotenuse from two legs.
  • ✓Find a missing leg.
Essential Question

How can you find the length of a ramp, a ladder, or the diagonal of a screen — without measuring it directly?

0
Lesson Phases
0
Vocabulary Terms
0
Standards Aligned
0
Interactive Task
Model · Interactive Tool

The Right Triangle Tool

Project this and change the two legs. Watch a² + b² add up to c², and see the hypotenuse calculated automatically.

📐 Change the legs — find the hypotenuseTry it
Leg a: 3
Leg b: 4

The Lesson · Gradual Release (I Do · We Do · You Do)

50 minutes, five moves

Tap any phase to open the teacher moves and student actions.

1

Launch — The Hidden Side

5 min

Find a ladder’s length without a tape measure — how?

👩‍🏫 Teacher Moves

  • Pose the ladder/ramp problem.
  • Ask how to find the slanted side.
  • Set the goal: the Pythagorean theorem.

🎒 Student Actions

  • Consider the problem.
  • Wonder how.
  • Predict the rule.
2

I Do — a² + b² = c²

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Name legs and hypotenuse.
  • State a² + b² = c².
  • Solve for the hypotenuse.

🎒 Student Actions

  • Name the sides.
  • Learn the rule.
  • Find c.
3

We Do — Explore Together

13 min

Class drives the Right Triangle Tool.

👩‍🏫 Teacher Moves

  • Change the legs; read c.
  • Verify a² + b² = c².
  • Estimate before computing.

🎒 Student Actions

  • Set the legs.
  • Read c.
  • Verify the rule.
4

You Do — On Your Own

12 min

Students apply the theorem.

👩‍🏫 Teacher Moves

  • Assign missing-side problems.
  • Find the hypotenuse and a leg.
  • Solve a real-world problem.

🎒 Student Actions

  • Find the hypotenuse.
  • Find a leg.
  • Solve a problem.
5

Close — Close

5 min

One triangle.

👩‍🏫 Teacher Moves

  • Give two legs; find c.
  • Show the work.
  • Hand out the exit ticket.

🎒 Student Actions

  • Find c.
  • Show a²+b²=c².
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics, including the Standards for Mathematical Practice.

CCSS
8.G.7

Apply the Pythagorean theorem to find unknown side lengths in right triangles.

CCSS
8.G.6

Explain a proof of the Pythagorean theorem and its converse.

CCSS
8.G.8

Apply the Pythagorean theorem to find the distance between two points.

SMP
MP.7

Look for and make use of structure in right triangles.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Label legs and hypotenuse.
  • Sentence frame: “a² + b² = c².”
  • Use a 3-4-5 triangle.

Support & Access

IEP / 504
  • Use whole-number (3-4-5) triangles.
  • Provide the formula on a card.
  • Use a calculator for square roots.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Find a missing leg.
  • Use it for the distance between two points.
  • Check if a triangle is right (the converse).
Materials

What to gather

  • 📽️Projector / board
  • 📓Math notebooks
  • 💻The Right Triangle Tool
  • 📐Rulers
  • 🧮Calculators
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

right trianglea triangle with a 90° anglelega side that forms the right anglehypotenusethe side opposite the right angle (longest)squarea number times itselfsquare rootthe reverse of squaringtheorema proven mathematical ruleright anglea 90° cornerdiagonala slanted line across a shape
Evaluate

Exit Ticket

Preview the three formative checks. Tap “Sample answer” to see what mastery looks like — hide them before you print for students.

QUESTION 1
State the Pythagorean theorem.
a² + b² = c² (in a right triangle, the squares of the legs add to the square of the hypotenuse).
QUESTION 2
If the legs are 3 and 4, what is the hypotenuse?
5 (because 9 + 16 = 25, and √25 = 5).
QUESTION 3
Which side is the hypotenuse?
The longest side — the one opposite the right angle.

Going deeper? Distance between points.

Have students use the Pythagorean theorem to find the distance between two points on a coordinate grid. A printable distance sheet is in the Math library.

Study · Flashcards

Study the key terms

Tap a card to flip it, then rate whether you knew it. Built from this lesson’s vocabulary.

🃏 The Pythagorean TheoremFlip
Card 1
Term
Tap to flip →
Meaning
0

Nice work!

Practice · Quiz

Check your understanding

A quick self-check with instant feedback, drawn from this lesson’s key terms.

📝 The Pythagorean TheoremQuiz
Score: 0
1 / 6
Question 1
0%

Nice work!

Practice · Worksheet

Printable worksheet

A print-and-go review sheet with a built-in answer key. Tap “Show answer key” to reveal answers, or print the clean version for students.

🖨️ The Pythagorean TheoremPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: diagonal, hypotenuse, leg, right angle, right triangle, square, square root, theorem
  1. a slanted line across a shape
  2. a number times itself
  3. the side opposite the right angle (longest)
  4. a proven mathematical rule
  5. a triangle with a 90° angle
  6. the reverse of squaring
  7. a 90° corner
  8. a side that forms the right angle

Part B · Show what you learned

  1. State the Pythagorean theorem.
  2. If the legs are 3 and 4, what is the hypotenuse?
  3. Which side is the hypotenuse?
Answer key — Part A: 1) diagonal · 2) square · 3) hypotenuse · 4) theorem · 5) right triangle · 6) square root · 7) right angle · 8) leg
Part B: 1) a² + b² = c² (in a right triangle, the squares of the legs add to the square of the hypotenuse). 2) 5 (because 9 + 16 = 25, and √25 = 5). 3) The longest side — the one opposite the right angle.