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.

Start hereWords for this lesson

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

Word	What it means	Example
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 examplesurface area of a square pyramid

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

A square pyramid has a base 10 cm on each side. Each triangular face has a slant height of 12 cm. Find the surface area.

The net is a square base with 4 congruent triangles, one on each side.
Area of the square base. 10 × 10 = 100
One triangular face uses the slant height. ½ × 10 × 12 = 60
Four faces plus the base. 60 × 4 = 240, then 240 + 100 = 340

Answer: The surface area is 340 square centimeters (340 cm²).

Now you trySame steps, your turn

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

A square pyramid has a base 14 in. on each side and a slant height of 9 in. Find the surface area.

Sketch the net. Square: Triangles:
Area of the square base. × = in²
One triangular face. ½ × × = in²
All the faces. × = , then + = in²

Answer:surface area of the pyramid

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 outA 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 — forgetting to add the base area at the end. This leaves the total surface area too small by exactly the base's area. Before you submit, ask: 'Did I add the base area to the four lateral faces, or did I stop after the sides?'

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.

Part 1Understand the ideamodels and meaning

1Complete the table. Show how you found each value.

Find the area of one lateral face for each pyramid.

Pyramid Base Edge Slant Height Area of One Face
Pyramid 1 4 in 9 in
Pyramid 2 6 cm 5 cm
Pyramid 3 10 ft 8 ft

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

Pyramid	Base Edge	Slant Height
Pyramid 1	4 in	9 in
Pyramid 2	6 cm	5 cm
Pyramid 3	10 ft	8 ft

Part 2Apply itreal-world problems

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?
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
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

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 ___.

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.

Part 1Understand the ideamodels and meaning

1Complete the table. Show how you found each value.

Find the surface area of each square pyramid.

Pyramid Base Edge Slant Height Base Area Total SA
Pyramid X 12 in 10 in
Pyramid Y 8 cm 6 cm
Pyramid Z 14 ft 11 ft

Show your work
2Use the model to solve. Show your work.

Total lateral area = ? in² (SA − base = 260 − 100).
Show your work
Answer:
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

Pyramid	Base Edge	Slant Height
Pyramid X	12 in	10 in
Pyramid Y	8 cm	6 cm
Pyramid Z	14 ft	11 ft

Part 2Apply itreal-world problems

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

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?

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.

Part 1Understand the ideamodels and meaning

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.

Shape Base Area Lateral/Side Area Total SA
Square pyramid (slant h = 8) 36
Rectangular prism (h = 8) 36 (× 2 bases)

Show your work

Shape	Base Area	Lateral/Side Area	Total SA
Square pyramid (slant h = 8)	36
Rectangular prism (h = 8)	36 (× 2 bases)

Part 2Explain and correctreasoning

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:
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?

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