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

Lesson 5.9Area of Regular Polygons

Set B · Version ASecond form · supported 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.

Why This Matters Your architecture firm is designing a hexagonal skylight for the lobby of a new museum. The glass fabricator needs to know the total area so they can cut the correct amount of glass. Each side of the hexagon is 6 feet, and the height of each triangle formed from the center is 5.2 feet. The height is rounded to the nearest tenth of a foot, so the area is approximate.

RememberArea of Regular Polygons

Area of Regular Polygons. Formula: Area = (Number of Sides) × (Area of One Triangle) A regular polygon is a closed shape with straight sides where every side is the same length and every angle is the same size. We can decompose it (break it apart) into equal triangles from the center, then add up their areas.

  1. Decompose the regular polygon from center into n congruent triangles.
  2. Find the area of one triangle using A = ½ × side × apothem (height).
  3. Multiply by the total number of sides

Words: Regular Polygon (Polígono regular)Decompose (Descomponer)Triangle (Triángulo)Composite (Compuesto)Formula (Fórmula)

  1. 1Circle the letter of the best answer. Show how you know.

    How many triangular faces does a square pyramid have?

    1. A3
    2. B4
    3. C5
    4. D6
    Show your work
  2. 2Circle the letter of the best answer. Show how you know.

    To find the surface area of a pyramid you add…

    1. Athe area of the base and the areas of all the triangular faces
    2. Bthe volume and the base area
    3. Cthe base perimeter and the height
    4. Donly the triangular faces
    Show your work
  3. 3Circle the letter of the best answer. Show how you know.

    A triangular pyramid has a base area of 20 in² and three faces of 14 in² each. What is its surface area?

    1. A34 in²
    2. B54 in²
    3. C62 in²
    4. D62 in³
    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. 4Circle the letter of the best answer. Show how you know.

    A square pyramid has a base 6 cm by 6 cm and four triangular faces of 15 cm² each. What is its surface area?

    1. A51 cm²
    2. B60 cm²
    3. C96 cm²
    4. D96 cm³
    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?
  5. 5Circle the letter of the best answer. Show how you know.

    The parks department hired your firm to finish a regular octagonal gazebo at the center of Patterson Park. Seen from above, the gazebo floor decomposes into 8 congruent triangles that meet at the center point, and each triangle has a base of 5 feet and a height of 6 feet. The flooring contractor charges $3 per square foot and needs a total price before Friday's budget meeting. The central-triangle height is rounded to the nearest foot. Estimate the area and price with that height. What is the area of one center triangle?

    1. A7.5 sq ft
    2. B11 sq ft
    3. C15 sq ft
    4. D30 sq 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?
  6. 6Circle the letter of the best answer. Show how you know.

    The parks department hired your firm to finish a regular octagonal gazebo at the center of Patterson Park. Seen from above, the gazebo floor decomposes into 8 congruent triangles that meet at the center point, and each triangle has a base of 5 feet and a height of 6 feet. The flooring contractor charges $3 per square foot and needs a total price before Friday's budget meeting. The central-triangle height is rounded to the nearest foot. Estimate the area and price with that height. What is the total floor area of the octagon?

    1. A15 sq ft
    2. B40 sq ft
    3. C120 sq ft
    4. D360 sq 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?

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.

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
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

Set B · Version BSecond form · core 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.

Why This Matters Your architecture firm is designing a hexagonal skylight for the lobby of a new museum. The glass fabricator needs to know the total area so they can cut the correct amount of glass. Each side of the hexagon is 6 feet, and the height of each triangle formed from the center is 5.2 feet. The height is rounded to the nearest tenth of a foot, so the area is approximate.

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

    Each tile decomposes into congruent triangles from its center. Work out the area of ONE triangle, then sort the tile. Use the given heights; nonsquare heights are rounded, so their areas are estimates.

    Groups: One triangle ≤ 12 sq ft One triangle > 12 sq ft
    • Square tile: base 4 ft, height 2 ft
    • Hexagon tile: base 4 ft, height ≈ 3.5 ft
    • Octagon tile: base 6 ft, height ≈ 7.2 ft
    • Pentagon tile: base 8 ft, height ≈ 5.5 ft
  2. 2Write the letter of the matching item on each line.

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

    1. Square tile: 4 triangles, each ½×9×4.5
    2. Octagon tile: 8 triangles, each about ½×3×3.6
    3. Hexagon tile: 6 triangles, each about ½×5×4.3
    4. Equilateral triangle tile: 1 whole triangle, about ½×6×5.2
    • A81 sq ft
    • BAbout 43.2 sq ft
    • CAbout 64.5 sq ft
    • DAbout 15.6 sq ft
Part 2Apply itreal-world problems
  1. 3Circle the letter of the best answer. Show how you know.

    The parks department hired your firm to finish a regular octagonal gazebo at the center of Patterson Park. Seen from above, the gazebo floor decomposes into 8 congruent triangles that meet at the center point, and each triangle has a base of 5 feet and a height of 6 feet. The flooring contractor charges $3 per square foot and needs a total price before Friday's budget meeting. The central-triangle height is rounded to the nearest foot. Estimate the area and price with that height. What will the flooring cost?

    1. A$45
    2. B$120
    3. C$360
    4. D$960
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
  2. 4Circle the letter of the best answer. Show how you know.

    A regular hexagonal floor tile splits from its center into 6 congruent triangles, each with an area of 12 cm². What is the total area of the tile?

    1. A12 cm²
    2. B18 cm²
    3. C36 cm²
    4. D72 cm²
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
  3. 5Circle the letter of the best answer. Show how you know.

    A regular pentagonal tile splits into 5 congruent triangles. Each triangle has a base of 4 in and a height of about 2.8 in. What is the total area of the tile? Use the given heights; nonsquare heights are rounded, so their areas are estimates.

    1. A5.6 in²
    2. B11.2 in²
    3. C28 in²
    4. D56 in²
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
  4. 6Circle the letter of the best answer. Show how you know.

    An octagon tile splits into equal triangles, each with base 6 ft and height ≈ 7.2 ft. How many triangles, and what is the total area? Use the given heights; nonsquare heights are rounded, so their areas are estimates.

    1. A21.6 sq ft
    2. B129.6 sq ft
    3. C151.2 sq ft
    4. D172.8 sq ft
    Formula
    Put the numbers in
    Work it out
    Answer with its unit

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?

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help