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Polygon Angle Sums: Interior Angles

Tenth graders discover the formula for the sum of a polygon’s interior angles — (n−2) × 180° — by splitting any polygon into triangles, and find each angle of a regular polygon.

Grade 10Polygons55 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
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Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓Find the interior angle sum.
  • ✓Use (n−2) × 180°.
  • ✓Find each angle of a regular polygon.
  • ✓Split polygons into triangles.
Essential Question

A triangle’s angles sum to 180°. A quadrilateral’s to 360°. Is there a pattern that predicts the angle sum of ANY polygon — a pentagon, a decagon, or beyond?

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

The Polygon Explorer

Project this and change the number of sides. Watch the polygon form — and see how splitting it into triangles gives the angle sum!

⬢ Explore polygon angle sumsTry it
Number of sides: 6

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

55 minutes, five moves

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

1

Launch — Is There a Pattern?

5 min

Triangle: 180°. Quadrilateral: 360°. What about a decagon?

👩‍🏫 Teacher Moves

  • Show angle sums for 3 and 4 sides.
  • Ask for a pattern.
  • Set the goal: angle sums.

🎒 Student Actions

  • See the sums.
  • Look for a pattern.
  • Predict a formula.
2

I Do — Split Into Triangles

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Split a polygon into triangles.
  • Count (n−2) triangles.
  • Multiply by 180°.

🎒 Student Actions

  • Split it.
  • Count triangles.
  • Multiply.
3

We Do — Explore Together

13 min

Class uses the Polygon Explorer.

👩‍🏫 Teacher Moves

  • Change the sides.
  • Read the angle sum.
  • Find each angle.

🎒 Student Actions

  • Change sides.
  • Read the sum.
  • Find each angle.
4

You Do — On Your Own

12 min

Students compute.

👩‍🏫 Teacher Moves

  • Find interior angle sums.
  • Find each regular-polygon angle.
  • Use the formula.

🎒 Student Actions

  • Find sums.
  • Find each angle.
  • Use the formula.
5

Close — Close

5 min

One polygon.

👩‍🏫 Teacher Moves

  • Give a polygon.
  • Find its angle sum.
  • Hand out the exit ticket.

🎒 Student Actions

  • Find the sum.
  • Explain it.
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics (High School Geometry).

CCSS
G-CO.13

Construct regular polygons inscribed in a circle.

CCSS
G-CO.10

Prove theorems about triangles, including the angle sum.

CCSS
G-MG.1

Use geometric shapes and their properties to model objects.

CCSS
G-CO.1

Know precise definitions of geometric figures.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Split-into-triangles diagram.
  • Sentence frame: “(n−2) × 180 = ___.”
  • Use polygon cutouts.

Support & Access

IEP / 504
  • Start with triangles and quadrilaterals.
  • Split polygons visually.
  • Provide the formula.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Find exterior angle sums.
  • Find n from a given angle.
  • Explore star polygons.
Materials

What to gather

  • 📽️Projector / board
  • 📐Polygon cutouts
  • 💻The Polygon Explorer
  • 🧮Calculators
  • ✏️Pencils
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

polygona closed figure of straight sidesinterior anglean angle inside the polygonangle sumthe total of all interior anglesregular polygonall sides and angles equalpentagona 5-sided polygonhexagona 6-sided polygonvertexa corner pointexterior anglean angle outside, on a straight line
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
What is the interior angle sum of a pentagon (5 sides)?
540° ((5−2) × 180).
QUESTION 2
What is each interior angle of a regular hexagon?
120° (720° ÷ 6).
QUESTION 3
What is the formula for the interior angle sum of an n-sided polygon?
(n − 2) × 180°.

Going deeper? Exterior angles.

Have students discover that the exterior angles of ANY polygon always sum to 360°. A printable polygon-angles 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.

🃏 Polygon Angle SumsFlip
Card 1
Term
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Meaning
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Practice · Quiz

Check your understanding

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

📝 Polygon Angle SumsQuiz
Score: 0
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Question 1
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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.

🖨️ Polygon Angle SumsPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: angle sum, exterior angle, hexagon, interior angle, pentagon, polygon, regular polygon, vertex
  1. an angle inside the polygon
  2. the total of all interior angles
  3. a 5-sided polygon
  4. a closed figure of straight sides
  5. a corner point
  6. all sides and angles equal
  7. an angle outside, on a straight line
  8. a 6-sided polygon

Part B · Show what you learned

  1. What is the interior angle sum of a pentagon (5 sides)?
  2. What is each interior angle of a regular hexagon?
  3. What is the formula for the interior angle sum of an n-sided polygon?
Answer key — Part A: 1) interior angle · 2) angle sum · 3) pentagon · 4) polygon · 5) vertex · 6) regular polygon · 7) exterior angle · 8) hexagon
Part B: 1) 540° ((5−2) × 180). 2) 120° (720° ÷ 6). 3) (n − 2) × 180°.