Grade 6 Mathematics · Unit 5: Solve Area, Surface Area, and Volume ProblemsStandard 6.GR.4

Lesson 5.7Determine Surface Area of Prisms

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can find the surface area of rectangular and triangular prisms.

Why This Matters Your class is painting the outside of different time capsule shapes to protect them from weather. Some capsules are rectangular prisms and others are triangular prisms (wedge-shaped). To buy the right amount of paint, you need to calculate the surface area of each shape! One capsule is a rectangular prism 7 inches long, 4 inches wide, and 3 inches tall. Another is a triangular prism whose triangle bases are 8 inches wide and 3 inches tall, with slant sides of 5 inches, and the prism is 9 inches long.

Start hereWords for this lesson

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

WordWhat it meansExample
Surface areaSpanish: Área de superficie The total area of all the flat sides of a solid. add every face
Rectangular prismSpanish: Prisma rectangular A solid box shape with six flat rectangle sides. 3 congruent pairs
Triangular prismSpanish: Prisma triangular A solid with two triangle ends and three flat rectangle sides. Shaped like a tent or a wedge of cheese — 2 triangle ends + 3 rectangle sides
FaceSpanish: Cara One flat side of a solid shape. top, side, end
NetSpanish: Plantilla (desarrollo plano) A flat shape that folds up into a solid. unfolded box
Lateral faceSpanish: Cara lateral The side faces of a solid, not the top or bottom. On a triangular prism, the 3 rectangles that wrap around the sides are lateral faces

How it worksWorked example

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

A closed box is 6 cm × 4 cm × 3 cm.

  1. Faces: 6 × 4 = 24, 6 × 3 = 18, 4 × 3 = 12.
  2. Double each: 48, 36, 24.
  3. Add: 48 + 36 + 24 = 108.

Answer: SA = 108 cm²

Now you trySame steps, your turn

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

Now a rectangular prism: 5 long, 4 wide, 3 tall. How many faces does it have?

Answer:

Say it and write itSentence starters

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

  • The prism has faces in matching pairs.
  • The three different face areas are , , and .
  • I doubled each face area because .
  • The surface area is square units.

Word bank surface areanetfacecongruentpairprismunfoldadddoublesquare unitstopside

Watch out!A common mistake

A common mistake in Surface Area of Prisms is confusing surface area with volume

Grade 6 Mathematics · Unit 5: Solve Area, Surface Area, and Volume ProblemsStandard 6.GR.4

Lesson 5.7Determine Surface Area of Prisms

Version ASupported practice

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

Learning target I can find the surface area of rectangular and triangular prisms.

Why This Matters Your class is painting the outside of different time capsule shapes to protect them from weather. Some capsules are rectangular prisms and others are triangular prisms (wedge-shaped). To buy the right amount of paint, you need to calculate the surface area of each shape! One capsule is a rectangular prism 7 inches long, 4 inches wide, and 3 inches tall. Another is a triangular prism whose triangle bases are 8 inches wide and 3 inches tall, with slant sides of 5 inches, and the prism is 9 inches long.

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

    Put the steps in order for finding the surface area of a triangular prism.

    Groups: Steps in order (first → last)
    • Find the area of one triangular base
    • Multiply the triangle area by 2 (two bases)
    • Find the area of each rectangular lateral face
    • Add all three rectangle areas together
    • Add the two triangle areas + all rectangle areas = SA

    Hint The cards describe finding a triangular prism's SA: triangles first, then rectangles, then a final total.

  2. 2Write the letter of the matching item on each line.

    Match each solid to its description.

    1. Rectangular prism
    2. Triangular prism
    3. Cube
    4. Surface area
    5. Volume
    6. Net
    • A6 faces (all rectangles)
    • B5 faces (2 triangles + 3 rectangles)
    • C6 identical square faces
    • DUses square units (in²)
    • EUses cubic units (in³)
    • F2D pattern that folds into 3D

    Hint Read each card: some describe face counts, some describe units, and one describes a flat fold-out pattern.

Part 2Apply itreal-world problems
  1. 3Circle the letter of the best answer. Show how you know.

    A time capsule is a rectangular prism with l = 5 cm, w = 4 cm, h = 3 cm. What surface area needs paint?

    1. A47 cm²
    2. B60 cm²
    3. C94 cm²
    4. D94 cm³

    Hint You need surface area — the total of all 6 face areas of the 5 × 4 × 3 prism — not the space inside.

    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. 4Circle the letter of the best answer. Show how you know.

    How many faces does a triangular prism have?

    1. A3
    2. B4
    3. C5
    4. D6

    Hint Picture a tent or a triangular chocolate box: two triangle ends plus rectangles wrapping around the sides.

    Show your work
  3. 5Circle the letter of the best answer. Show how you know.

    A wedge-shaped time capsule is a triangular prism. It has two triangular bases each with area 12 cm² and three rectangular faces with areas 30 cm², 40 cm², and 40 cm². What is the total surface area to paint?

    1. A110 cm²
    2. B122 cm²
    3. C134 cm²
    4. D134 cm³

    Hint All five face areas are given: two triangles of 12 cm² each, and rectangles of 30, 40, and 40 cm².

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

    Spot the Common Mistake — A Forgotten Face

    1. 1DimensionsRectangular prism time capsule: l = 5 in, w = 3 in, h = 2 in.
    2. 2FormulaSA = 2lw + 2lh + 2wh (add all 6 faces).
    3. 3CalculateThe student wrote SA = 2(5×3) + 2(5×2) + (3×2) = 30 + 20 + 6 = 56 in².

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

    Hint A prism has 6 faces, in 3 matching pairs — every pair gets multiplied by 2.

    Correct work
    Correct answer:

Explain your thinkingCompare Capsule A (rectangular, SA = 248 in²) and Capsule C (triangular, SA = 184 in²). Even though they are similar in size, why does the rectangular prism need more paint?

Sentence starter The rectangular prism has ___ faces while the triangular prism has ___ faces. The rectangular prism's SA is ___ in² which is ___ more than the triangular prism's ___ in² because ___.

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.4

Lesson 5.7Determine Surface Area of Prisms

Version BCore practice

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

Learning target I can find the surface area of rectangular and triangular prisms.

Why This Matters Your class is painting the outside of different time capsule shapes to protect them from weather. Some capsules are rectangular prisms and others are triangular prisms (wedge-shaped). To buy the right amount of paint, you need to calculate the surface area of each shape! One capsule is a rectangular prism 7 inches long, 4 inches wide, and 3 inches tall. Another is a triangular prism whose triangle bases are 8 inches wide and 3 inches tall, with slant sides of 5 inches, and the prism is 9 inches long.

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

    Calculate the surface area of each rectangular prism time capsule.

    CapsuleLengthWidthHeightSurface Area
    Capsule E12 in8 in5 in
    Capsule F9 cm6 cm6 cm
    Capsule G15 ft4 ft3 ft
    Show your work
  2. 2Use the model to solve. Show your work.
    Total SA: 340 cm²Top & BottomFront & BackLeft & Right

    What is the combined area of the left and right faces?

    Show your work
    Answer:
  3. 3Write the name of the correct group on each line.

    Sort these prisms from LEAST surface area to GREATEST.

    Groups: Least → Greatest surface area
    • Cube 3×3×3: SA = 54
    • Rect prism 5×4×3: SA = 94
    • Rect prism 8×5×3: SA = 158
    • Rect prism 10×6×4: SA = 248
Part 2Apply itreal-world problems
  1. 4Circle the letter of the best answer. Show how you know.

    A wedge-shaped time capsule is a triangular prism. Its triangular bases have base 6 in and height 4 in. The prism is 10 in long and the two slant sides of the triangle are each 5 in. What surface area needs paint?

    1. A124 in²
    2. B160 in²
    3. C184 in²
    4. D184 in³
    Formula
    Put the numbers in
    Work it out
    Answer with its unit
Part 3Explain and correctreasoning
  1. 5Find the mistake. Explain it, then show the correct work.

    Spot the Common Mistake — Surface Area or Volume?

    1. 1DimensionsRectangular prism time capsule: l = 7 in, w = 4 in, h = 3 in. Find the surface area.
    2. 2FormulaSurface area adds all 6 faces: SA = 2lw + 2lh + 2wh.
    3. 3CalculateThe student multiplied the three dimensions: 7 × 4 × 3 = 84, so they wrote SA = 84 in².

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

    Correct work
    Correct answer:

Explain your thinkingCompare Capsule A (rectangular, SA = 248 in²) and Capsule C (triangular, SA = 184 in²). Even though they are similar in size, why does the rectangular prism need more paint?

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.4

Lesson 5.7Determine Surface Area of Prisms

ChallengeExtension

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

Learning target I can find the surface area of rectangular and triangular prisms.

Why This Matters Your class is painting the outside of different time capsule shapes to protect them from weather. Some capsules are rectangular prisms and others are triangular prisms (wedge-shaped). To buy the right amount of paint, you need to calculate the surface area of each shape! One capsule is a rectangular prism 7 inches long, 4 inches wide, and 3 inches tall. Another is a triangular prism whose triangle bases are 8 inches wide and 3 inches tall, with slant sides of 5 inches, and the prism is 9 inches long.

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

    A rectangular prism and a triangular prism have similar dimensions. Compare their surface areas and volumes.

    ShapeFacesSurface AreaVolume
    Rect prism: 6×4×36
    Tri prism: base=6, h=4, length=3, sides=55
    Show your work
Part 2Explain and correctreasoning
  1. 2Find the mistake. Explain it, then show the correct work.

    Find the Surface Area Error

    1. 1DimensionsRectangular prism: l = 12 cm, w = 7 cm, h = 5 cm
    2. 2FormulaSA = 2lw + 2lh + 2wh
    3. 3CalculateSA = 2(12×7) + 2(12×5) + 2(7×5) = 168 + 120 + 35 = 323 cm²
    4. 4AnswerSurface Area = 323 cm²

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

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

    Compare two closed containers: a rectangular prism (8 × 6 × 4 in) and a triangular prism whose ends are right triangles with perpendicular legs 6 in and 8 in and hypotenuse 10 in. The triangular prism is 8 in long. Which uses less protective film to cover every face? Show your work.

    Write your answer

Explain your thinkingCompare Capsule A (rectangular, SA = 248 in²) and Capsule C (triangular, SA = 184 in²). Even though they are similar in size, why does the rectangular prism need more paint?

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