Grade 6 Mathematics · Unit 5: Solve Area, Surface Area, and Volume ProblemsStandard 6.GR.1

Lesson 5.9Area of Regular Polygons

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can find the area of a regular polygon by decomposing it into triangles.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
Regular PolygonSpanish: Polígono regular A closed flat shape made of straight sides, where every side is the same length and every angle is the same size. A stop sign is a regular octagon — all 8 sides are the same length and all 8 angles are equal
DecomposeSpanish: Descomponer To break a shape into smaller, simpler shapes. Draw lines from the center of a hexagon to each corner → you get 6 equal triangles you can find the area of
TriangleSpanish: Triángulo A shape with three sides. Each triangle inside a decomposed hexagon has a base = one side of the hexagon and height = distance from center to side
CompositeSpanish: Compuesto Made by putting two or more simple shapes together. 6 triangles, each with area 15 sq ft, combine to form a hexagon with total area = 6 × 15 = 90 sq ft
FormulaSpanish: Fórmula A math rule written with symbols. For a regular hexagon: Total Area = 6 × (½ × b × h), where b = side length and h = height of each triangle

How it worksWorked example

These numbers are not on your problems. The steps are. Follow them with your own numbers.

My hexagon has 6 equal sides. Each side is 6 feet, and each center triangle has a height of 5.2 feet. Use the rounded height to estimate the area.

  1. First I find one triangle: A = ½ × base × height = ½ × 6 × 5.2.
  2. I multiply: 6 × 5.2 = 31.2, then ½ × 31.2 = 15.6 square feet.
  3. A hexagon has 6 sides, so it has 6 equal triangles.
  4. I multiply: 6 × 15.6 = 93.6. So the total area is 93.6 square feet.

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

Now a smaller regular pentagon: each of its 5 triangles has base 4 cm and height 3 cm. Use the rounded height to estimate the area.

Answer:

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

  • The total area is because one triangle is and there are triangles.

Word bank Regular PolygonDecomposeTriangleCompositeFormulatrianglehalfbaseheightmultiplytotalnumber of sides

Watch outA common mistake

A common mistake in Area of Regular Polygons is stopping after finding the area of ONE triangle and reporting that as the total area — or adding the number of triangles instead of multiplying. Remember: total area = (one triangle's area) × (number of sides), not (one triangle's area) + (number of sides).

Grade 6 Mathematics · Unit 5: Solve Area, Surface Area, and Volume ProblemsStandard 6.GR.1

Lesson 5.9Area of Regular Polygons

Version ASupported practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can find the area of a regular polygon by decomposing it into triangles.

Part 1Understand the ideamodels and meaning
  1. 1Write the name of the correct group on each line.

    Each regular polygon below is cut into congruent triangles by drawing a line from the center to every corner. Sort by how many triangles it makes.

    Groups: More than 5 triangles 5 or fewer triangles
    • Regular hexagon
    • Regular octagon
    • Equilateral triangle
    • Square
    • Regular pentagon

    Hint Picture it: one spoke from the center to each corner. Each triangle sits on one side.

Part 2Apply itreal-world problems
  1. 2Circle the letter of the best answer. Show how you know.

    A regular hexagon is divided into 6 equal triangles from the center. Each triangle has a base of 6 ft and a height of 5.2 ft. What is the area of one triangle? Use the given heights; nonsquare heights are rounded, so their areas are estimates.

    1. A11.2 sq ft
    2. B15.6 sq ft
    3. C15.6 ft
    4. D31.2 sq ft

    Hint Focus on just ONE of the 6 triangles: its base is 6 ft and its height is 5.2 ft.

    Formula
    Put the numbers in
    Work it out
    Answer with its unit
    1. 1Name itWhich measure is asked?
    2. 2Write the formulaBefore any numbers.
    3. 3SubstituteMatch each letter.
    4. 4Unitunits, sq units, or cubic?
  2. 3Circle the letter of the best answer. Show how you know.

    A regular hexagonal skylight has 6 central triangles, each with base 6 ft and rounded height 5.2 ft. Using those measurements, what is the approximate total area?

    1. A15.6 sq ft
    2. B62.4 sq ft
    3. C93.6 sq ft
    4. D187.2 sq ft

    Hint Find one central triangle’s area: ½ × 6 × 5.2 = 15.6 sq ft using the rounded height.

    Formula
    Put the numbers in
    Work it out
    Answer with its unit
    1. 1Name itWhich measure is asked?
    2. 2Write the formulaBefore any numbers.
    3. 3SubstituteMatch each letter.
    4. 4Unitunits, sq units, or cubic?
  3. 4Circle the letter of the best answer. Show how you know.

    A regular hexagon is split into 6 triangles, each with base 4 cm and height 3.5 cm. What is the total area? Use the given heights; nonsquare heights are rounded, so their areas are estimates.

    1. A14 sq cm
    2. B21 sq cm
    3. C42 sq cm
    4. D84 sq cm

    Hint Two stages hide here: first one triangle (base 4 cm, height 3.5 cm), then all 6 of them.

    Formula
    Put the numbers in
    Work it out
    Answer with its unit
    1. 1Name itWhich measure is asked?
    2. 2Write the formulaBefore any numbers.
    3. 3SubstituteMatch each letter.
    4. 4Unitunits, sq units, or cubic?
  4. 5Circle the letter of the best answer. Show how you know.

    How many triangles can you make from the center of a regular pentagon?

    1. A3
    2. B5
    3. C6
    4. D10

    Hint 'Penta' is a clue about how many sides a pentagon has.

    Show your work
Part 3Explain and correctreasoning
  1. 6Find the mistake. Explain it, then show the correct work.

    Spot the Common Mistake (use rounded heights)

    1. 1Decompose the octagonA regular octagon = 8 congruent triangles
    2. 2Identify one trianglebase = 3 ft, height = 4 ft
    3. 3Find the area of one triangleA = ½ × 3 × 4 = 6 sq ft
    4. 4Find total areaTotal area = 5 × 6 = 30 sq ft

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Hint Look at Step 4. How many sides does an octagon actually have?

    Correct work
    Correct answer:

Explain your thinkingThe hexagon and the octagon both have triangles, but the totals are different. What two things decide the total area of a regular polygon?

Sentence starter The total area is ___ because one triangle is ___ and there are ___ triangles.

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 5: Solve Area, Surface Area, and Volume ProblemsStandard 6.GR.1

Lesson 5.9Area of Regular Polygons

Version BCore practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can find the area of a regular polygon by decomposing it into triangles.

Part 1Understand the ideamodels and meaning
  1. 1Complete the table. Show how you found each value.

    Complete the table: find one central triangle’s area and the polygon’s total area. Use the given rounded heights for estimates; the square dimensions are exact.

    PolygonNumber of TrianglesTriangle BaseTriangle HeightArea of One TriangleTotal Area
    Hexagon66 ft5.2 ft
    Pentagon58 ft5.5 ft
    Square45 ft2.5 ft
    Octagon84 ft4.8 ft
    Show your work
  2. 2Write the letter of the matching item on each line.

    Match each polygon to its total area. Use the given heights; nonsquare heights are rounded, so their areas are estimates.

    1. Hexagon: 6 triangles, each ½×6×5.2
    2. Pentagon: 5 triangles, each ½×8×5.5
    3. Octagon: 8 triangles, each ½×4×4.8
    4. Square: 4 triangles, each ½×5×2.5
    5. Decagon: 10 triangles, each about ½×3×4.6
    6. Equilateral triangle: 3 central triangles, each about ½×4×1.2
    • A93.6 sq ft
    • B110 sq ft
    • C76.8 sq ft
    • D25 sq ft
    • E69 sq ft
    • F7.2 sq ft
  3. 3Write the name of the correct group on each line.

    Find the TOTAL area of each regular polygon, then sort it against the 90 sq ft cutoff. Use the given heights; nonsquare heights are rounded, so their areas are estimates.

    Groups: Total area ≥ 90 sq ft Total area < 90 sq ft
    • Pentagon: 5 triangles, base 8 ft, height 5.5 ft
    • Hexagon: 6 triangles, base 6 ft, height 5.2 ft
    • Octagon: 8 triangles, base 4 ft, height 4.8 ft
    • Square: 4 triangles, base 5 ft, height 2.5 ft
  4. 4Mark and label each value on the number line.

    Find the central-triangle height: square area 100 sq ft, 4 triangles, base 10 ft

    012345678910
    Show your thinking
Part 2Explain and correctreasoning
  1. 5Find the mistake. Explain it, then show the correct work.

    Spot the Common Mistake (use rounded heights)

    1. 1Decompose the pentagonA regular pentagon = 5 congruent triangles
    2. 2Identify one trianglebase = 6 ft, height = 4 ft
    3. 3Find the area of one triangleA = 6 × 4 = 24 sq ft
    4. 4Find total areaTotal area = 5 × 24 = 120 sq ft

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:

Explain your thinkingThe hexagon and the octagon both have triangles, but the totals are different. What two things decide the total area of a regular polygon?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 5: Solve Area, Surface Area, and Volume ProblemsStandard 6.GR.1

Lesson 5.9Area of Regular Polygons

ChallengeExtension

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can find the area of a regular polygon by decomposing it into triangles.

Part 1Understand the ideamodels and meaning
  1. 1Complete the table. Show how you found each value.

    Use total area = number of triangles × ½ × base × height to find each missing value. Nonsquare measurements are rounded; use the displayed values.

    Regular PolygonNumber of TrianglesTriangle BaseTriangle HeightTotal Area
    Hexagon69 ft216 sq ft
    Square46 ft144 sq ft
    Octagon4 ft5 ft80 sq ft
    Decagon103 ft69 sq ft
    Show your work
Part 2Explain and correctreasoning
  1. 2Find the mistake. Explain it, then show the correct work.

    Find the Area Error (use rounded heights)

    1. 1Decompose the hexagonA regular hexagon = 6 triangles
    2. 2Find area of one triangleb = 5 m, h = 4.3 m
    3. 3Calculate one triangleA = ½ × 5 × 4.3 = 10.75 sq m
    4. 4Find total areaTotal = 10.75 + 6 = 16.75 sq m

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:
  2. 3Answer in complete sentences.

    A regular hexagon and a regular pentagon are each divided into congruent central triangles. In both polygons, each central triangle has area 15 sq ft. Which polygon has greater total area? Explain why the triangle count matters. The two polygons have different side lengths.

    Write your answer

Explain your thinkingThe hexagon and the octagon both have triangles, but the totals are different. What two things decide the total area of a regular polygon?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help