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

Level 0Most support · every hint printed

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.

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

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
    Steps to try
    1. Each row asks for the area of ONE triangular face, using its base edge and slant height.
    2. Use ½ × base edge × slant height for each row — multiply the two numbers, then halve.
    3. Pyramid 1: multiply 4 × 9 = 36, then take half. Write the result with in².
    Show your work
Part 2Apply itreal-world problems
  1. 2Circle 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
    Steps to try
    1. A square pyramid has TWO kinds of faces: one square base (5 × 5) and four identical triangles with slant height 7.
    2. SA = base area + 4 triangle areas, where each triangle is ½ × base edge × slant height.
    3. Base first: 5 × 5 = 25 in². Then one triangle: ½ × 5 × 7. Multiply that by 4 and add the base area.
    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.

    How many lateral (triangular) faces does a square pyramid have?

    1. A3
    2. B4
    3. C5
    4. D6
    Steps to try
    1. Lateral faces are the slanted triangles on the sides — the square base itself doesn't count.
    2. Each side of the square base holds up exactly one triangle. Count the sides of a square.
    3. Trace the base: it's a square. Count one triangle leaning against each side of that square.
    Show your work
  3. 4Circle 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
    Steps to try
    1. You have one triangle: base 8 in, slant height 6 in. Triangle area is not just base × height.
    2. Use area = ½ × base × height — multiply the two numbers first, then take half.
    3. Multiply 8 × 6 = 48 first. Then take half of that result.
    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 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