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

Lesson 5.8Determine Surface Area of Pyramids

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can find the surface area of a pyramid by adding the base area and the lateral faces.

Why This Matters Your class is building a pyramid-shaped display to showcase the time capsule at the school entrance. The pyramid will be covered in gold leaf to make it shine. To figure out how much gold leaf you need, you must calculate the total surface area — the base plus all the triangular faces! The pyramid has a square base that is 10 inches on each side, and each triangular face has a slant height of 12 inches.

Start hereWords for this lesson

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

WordWhat it meansExample
PyramidSpanish: Pirámide A solid with a flat bottom and triangle sides that meet at one point on top. square base + 4 triangles
Slant heightSpanish: Altura inclinada (apotema) The height of a side triangle, measured along its slanted face. used in ½ × b × slant
Lateral faceSpanish: Cara lateral A triangle side of a pyramid, not the bottom. 4 on a square pyramid
BaseSpanish: Base The flat bottom of a pyramid. the square on the net
ApexSpanish: Ápice The point at the top of a pyramid where the sides meet. tip of the pyramid
Lateral areaSpanish: Área lateral The total area of just the side triangles, not the bottom. For a square pyramid: lateral area = 4 × (½ × base edge × slant height)

How it worksWorked example

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

Square base 10 cm; slant height 7 cm.

  1. Base: 10 × 10 = 100 cm².
  2. One triangle: ½ × 10 × 7 = 35 cm².
  3. Four triangles: 140. Add: 100 + 140.

Answer: SA = 240 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 smaller square pyramid: base 4 × 4, slant height 5. What is the base area?

Answer:

Say it and write itSentence starters

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

  • The base is a with an area of .
  • Each triangular face has a base of and a slant height of .
  • There are triangular faces, so together they cover .
  • The surface area is because I added .

Word bank pyramidbaseslant heightlateral faceapextrianglesquarecongruentaddsurface areasquare unitsnet

Watch out!A common mistake

A common mistake in Surface Area of Pyramids is finding the four triangular lateral faces correctly (½ × base edge × slant height, times 4) and then stopping

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

Lesson 5.8Determine Surface Area of Pyramids

Version ASupported practice

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

Learning target I can find the surface area of a pyramid by adding the base area and the lateral faces.

Why This Matters Your class is building a pyramid-shaped display to showcase the time capsule at the school entrance. The pyramid will be covered in gold leaf to make it shine. To figure out how much gold leaf you need, you must calculate the total surface area — the base plus all the triangular faces! The pyramid has a square base that is 10 inches on each side, and each triangular face has a slant height of 12 inches.

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

    Find the area of one lateral face for each pyramid.

    PyramidBase EdgeSlant HeightArea of One Face
    Pyramid 14 in9 in
    Pyramid 26 cm5 cm
    Pyramid 310 ft8 ft

    Hint Each row asks for the area of ONE triangular face, using its base edge and slant height.

    Show your work
  2. 2Write the name of the correct group on each line.

    Put the steps in order for finding the surface area of a square pyramid (base edge 6, slant height 10).

    Groups:
    • Find base area: 6 × 6 = 36
    • Find one lateral face: ½ × 6 × 10 = 30
    • Multiply by 4 faces: 4 × 30 = 120
    • Add base + lateral: 36 + 120
    • SA = 156 in²

    Hint The cards walk from base area to lateral faces to the final total for a base-6, slant-10 pyramid.

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

    A square pyramid has a base edge of 5 in and a slant height of 7 in. What is the surface area?

    1. A70 in²
    2. B95 in²
    3. C95 in³
    4. D120 in²

    Hint A square pyramid has TWO kinds of faces: one square base (5 × 5) and four identical triangles with slant height 7.

    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 lateral (triangular) faces does a square pyramid have?

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

    Hint Lateral faces are the slanted triangles on the sides — the square base itself doesn't count.

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

    What is the area of one triangular face with base 8 in and slant height 6 in?

    1. A14 in²
    2. B24 in²
    3. C24 in³
    4. D48 in²

    Hint You have one triangle: base 8 in, slant height 6 in. Triangle area is not just base × 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?
Part 3Explain and correctreasoning
  1. 6Find the mistake. Explain it, then show the correct work.

    Don't Forget the Base

    1. 1BoxSquare pyramid: base edge = 5 in, slant height = 6 in
    2. 2One Triangular Face½ × 5 × 6 = 15 in²
    3. 3Four Faces4 × 15 = 60 in²
    4. 4AnswerSA = 60 in² (stopped after the four triangular faces)

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

    Hint Did the student include the bottom of the pyramid?

    Correct work
    Correct answer:

Explain your thinkingDisplay B has the largest surface area (280 in²). What contributes more to the total — the base or the lateral faces? Why?

Sentence starter For Display B, the base area is ___ in² and the total lateral area is 4 × ___ = ___ in². The ___ contribute(s) more 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 5.8Determine Surface Area of Pyramids

Version BCore practice

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

Learning target I can find the surface area of a pyramid by adding the base area and the lateral faces.

Why This Matters Your class is building a pyramid-shaped display to showcase the time capsule at the school entrance. The pyramid will be covered in gold leaf to make it shine. To figure out how much gold leaf you need, you must calculate the total surface area — the base plus all the triangular faces! The pyramid has a square base that is 10 inches on each side, and each triangular face has a slant height of 12 inches.

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

    Find the surface area of each square pyramid.

    PyramidBase EdgeSlant HeightBase AreaTotal SA
    Pyramid X12 in10 in
    Pyramid Y8 cm6 cm
    Pyramid Z14 ft11 ft
    Show your work
  2. 2Use the model to solve. Show your work.
    Total SA: 260 in²BaseLateral Faces

    Total lateral area = ? in² (SA − base = 260 − 100).

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

    Sort these pyramids from LEAST surface area to GREATEST.

    Groups:
    • Base 4×4, slant 5
    • Base 6×6, slant 5
    • Base 6×6, slant 8
    • Base 10×10, slant 9
Part 2Apply itreal-world problems
  1. 4Circle the letter of the best answer. Show how you know.

    A square pyramid has a base area of 49 cm² and a total lateral area of 84 cm². What is the surface area?

    1. A49 cm²
    2. B84 cm²
    3. C133 cm²
    4. D133 cm³
    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.

    Did You Halve the Triangle?

    1. 1BoxSquare pyramid: base edge = 8 in, slant height = 10 in
    2. 2Base Area8 × 8 = 64 in²
    3. 3One Triangular Face8 × 10 = 80 in² (forgot the ½)
    4. 4AnswerSA = 64 + 4 × 80 = 64 + 320 = 384 in²

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

    Correct work
    Correct answer:

Explain your thinkingDisplay B has the largest surface area (280 in²). What contributes more to the total — the base or the lateral faces? Why?

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 5.8Determine Surface Area of Pyramids

ChallengeExtension

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

Learning target I can find the surface area of a pyramid by adding the base area and the lateral faces.

Why This Matters Your class is building a pyramid-shaped display to showcase the time capsule at the school entrance. The pyramid will be covered in gold leaf to make it shine. To figure out how much gold leaf you need, you must calculate the total surface area — the base plus all the triangular faces! The pyramid has a square base that is 10 inches on each side, and each triangular face has a slant height of 12 inches.

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

    Compare a square pyramid and a rectangular prism that sit on the same 6 × 6 base.

    ShapeBase AreaLateral/Side AreaTotal SA
    Square pyramid (slant h = 8)36
    Rectangular prism (h = 8)36 (× 2 bases)
    Show your work
Part 2Explain and correctreasoning
  1. 2Find the mistake. Explain it, then show the correct work.

    Find the Surface Area Error

    1. 1DimensionsSquare pyramid: base edge = 10 in, slant height = 8 in
    2. 2Base Area10 × 10 = 100 in²
    3. 3Lateral Area½ × 10 × 8 = 40 in² (one face). Total lateral = 40 × 3 = 120 in²
    4. 4AnswerSA = 100 + 120 = 220 in²

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

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

    Two pyramids have the same base area (64 in²) but different slant heights. Pyramid A has slant height 6 in and Pyramid B has slant height 10 in. How much more surface area does Pyramid B have? Why does slant height affect surface area but NOT base area?

    Write your answer

Explain your thinkingDisplay B has the largest surface area (280 in²). What contributes more to the total — the base or the lateral faces? Why?

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