Unit 10 · Standard 6.G.4
Surface Area of Pyramids
Key Vocabulary Level 1 support
Picture first, then the word, then a plain-language meaning. Say each word out loud.
Like the Great Pyramid of Giza — a square on the bottom, 4 triangles slanting up to a point
Pyramid
A solid with a flat bottom and triangle sides that meet at one point on top.
If you slide your finger from the bottom edge up the triangle face to the top point — that distance is the slant height
Slant height
The height of a side triangle, measured along its slanted face.
A square pyramid has 4 lateral faces — one triangle for each side of the square base
Lateral face
A triangle side of a pyramid, not the bottom.
A square pyramid sits on a square base; base area = side × side
Base
The flat bottom of a pyramid.
The pointy top of a pyramid — all the slanted edges connect here
Apex
The point at the top of a pyramid where the sides meet.
For a square pyramid: lateral area = 4 × (½ × base edge × slant height)
Lateral area
The total area of just the side triangles, not the bottom.
Key Ideas & Notes
- 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!
- Calculate the surface area of each pyramid display. SA = Base Area + Lateral Area. For each triangular face, use A = ½ × base × slant height.
Think About It
- How many faces does a square pyramid have?
- What shape is the base? What shape are the side faces?
- What is the difference between the height of the pyramid and the slant height?
My Notes
Guided Examples
Example 1
A square pyramid has a base edge of 5 in and a slant height of 7 in. What is the surface area?
Solution: Base: 5 × 5 = 25 in². Each lateral face: ½ × 5 × 7 = 17.5 in². Four faces: 4 × 17.5 = 70 in². SA = 25 + 70 = 95 in².
Answer: A. 95 in²
Example 2
How many lateral (triangular) faces does a square pyramid have?
Solution: A square pyramid has a square base and 4 triangular lateral faces — one for each side of the square.
Answer: A. 4
Example 3
What is the area of one triangular face with base 8 in and slant height 6 in?
Solution: Area of a triangle = ½ × base × height = ½ × 8 × 6 = 24 in².
Answer: A. 24 in²
Write About the Math The Writing Revolution
I can explain my work using the words pyramid, slant height, lateral face, and base.
1. Kernel Sentence subject + verb
Model: Pyramid is a solid with a flat bottom and triangle sides that meet at one point on top.Pirámide es un sólido con un fondo plano y lados triangulares que se unen en un punto arriba.
Write a kernel sentence about pyramid. Use a subject and a verb.Escribe una oración base sobre pirámide. Usa un sujeto y un verbo.
2. Sentence Expansion because · but · so
Kernel: Pyramid matters in mathPirámide importa en matemáticas
Expand the kernel three ways. Add a reason, a contrast, and a result.
Pyramid matters in math because ___.Pirámide importa en matemáticas porque ___.
Pyramid matters in math, but ___.Pirámide importa en matemáticas, pero ___.
Pyramid matters in math, so ___.Pirámide importa en matemáticas, entonces ___.
3. Sentence Types 4 ways to write a math idea
Tell one true fact about pyramid.Di un hecho verdadero sobre pyramid.
Pyramid ___.
Ask a question about pyramid.Haz una pregunta sobre pyramid.
How does ___ ?¿Cómo ___ ?
Show excitement about pyramid.Muestra entusiasmo sobre pyramid.
Wow, ___ !¡Guau, ___ !
Tell a partner what to do with pyramid.Dile a un compañero qué hacer con pyramid.
First, ___ .Primero, ___ .
4. Explain Your Reasoning use a sentence starter
The pyramid has ___ triangular faces.La pirámide tiene ___ caras triangulares.
I added the base and the ___.Sumé la base y las ___.
I see pyramids in ___.Veo pirámides en ___.
Try It
Solve on your own. Check the answer key when you are done.
1. What is the area of one triangular face with base 8 in and slant height 6 in?
- 24 in²
- 48 in²
- 14 in²
- 24 in³
2. A square pyramid has a base area of 49 cm² and a total lateral area of 84 cm². What is the surface area?
- 133 cm²
- 84 cm²
- 49 cm²
- 133 cm³
Stretch Your Thinking Level 2 enrichment
Challenge task — explain your reasoning in full 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?
Sentence starter: Pyramid A: SA = 64 + 4(½ × 8 × 6) = 64 + ___ = ___ in². Pyramid B: SA = 64 + 4(½ × 8 × 10) = 64 + ___ = ___ in². Pyramid B has ___ more in² because ___. Slant height only affects ___ because ___.
Reflect — Exit Ticket
A square pyramid has a base edge of 6 in and a slant height of 5 in. What is the total surface area?
- 96 in²
- 60 in²
- 96 in³
- 36 in²
Answer Key & Teacher Guide
- Try It 1: A. 24 in² — Area of a triangle = ½ × base × height = ½ × 8 × 6 = 24 in².
- Try It 2: A. 133 cm² — SA = Base Area + Lateral Area = 49 + 84 = 133 cm².
- Exit Ticket: A. 96 in² — Base: 6 × 6 = 36 in². Each lateral face: ½ × 6 × 5 = 15 in². Four faces: 4 × 15 = 60 in². SA = 36 + 60 = 96 in². Surface area uses square units (in²).
Writing (TWR) — what to look for
- Kernel sentence: A complete sentence needs a subject and a verb. Example: Pyramid is a solid with a flat bottom and triangle sides that meet at one point on top.
- Expansion: because gives a reason, but shows a contrast or exception, so shows a result. Answers vary; each must keep the kernel idea and add the correct kind of detail.
- Sentence types: Statement ends with a period, question with "?", exclamation with "!", and a command starts with an action verb (a "bossy" verb).