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Unit 5 · Standard 6.GR.4

Determine Surface Area of Pyramids

Lesson 5-8 · Learn It — how the math works

Why we're learning this

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.

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How do we find the surface area of a pyramid?

Surface area of a pyramid is the base area plus all the triangular side faces. A square pyramid has 1 square base and 4 triangular faces. Each triangle uses the slant height: ½ × base × slant height.

Picture itSee what the math looks like before you read about it

The blueprint shows a right square pyramid with the measurements of its base and its slant height labeled.
Where you see this in real lifeThe blueprint shows a right square pyramid with the measurements of its base and its slant height labeled.

Watch me solve itI do — read one step at a time

The problem looks like this
One triangular face of the pyramid: base 6 in., slant height 8 in.6 in.8 in.
  1. My square has a 6 in. × 6 in. and a of 8 in.6 in. × 6 in. base
  2. First the area:6 × 6 = 36
  3. Now one triangular face: ½ × 6 × 8 = 24. A square has 4 of them, so 4 × 24 = 96.
  4. I add the and the side faces:36 + 96 = 132
  5. So the total is 132 in². I used the (8) for the triangles, not the 's height.

Watch out — mistakes students make here

  • Add the base area too — total surface is base plus 4 triangles.
  • Surface area is square units (in²), not cubic.
  • Check the units — surface area is measured in square units, not cubic.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. Now a smaller square : 4 × 4, 5. What is the base area?base 4 × 4
  2. Right, 4 × 4 = 16. Now one triangular face: ½ × 4 × 5. What do you get?4 × 4 = 16
  3. Yes, 10. Four faces is 4 × 10 = 40. Now add the : 16 + 40. What is the ?4 × 10 = 40
  4. Yes, SA = 56 in². Don't forget to add the area!

On your ownYou do — solve this one by yourself

Before you start, remember:

  • In the table below, find the area of one , then the total of each display.
  • Remember: total SA = area + all the triangular faces.

Solve: Find the of each square .

PyramidBase EdgeSlant HeightBase AreaTotal SA
Pyramid X12 in10 in
Pyramid Y8 cm6 cm
Pyramid Z14 ft11 ft

Stuck? Open one hint at a time.

Hint 1

Each row needs two answers: the square base's area, then the TOTAL surface area (base + 4 triangles).

Hint 2

Base area = edge × edge. Then add 4 × (½ × edge × slant height) for the four triangular faces.

Hint 3

Pyramid X: base area is 12 × 12. Each triangle is ½ × 12 × 10 — find that, multiply by 4, and add to the base.

Check my answer
✅ Answers:
  • Pyramid X → Base Area: 144 in²
  • Pyramid X → Total SA: 384 in²
  • Pyramid Y → Base Area: 64 cm²
  • Pyramid Y → Total SA: 160 cm²
  • Pyramid Z → Base Area: 196 ft²
  • Pyramid Z → Total SA: 504 ft²
Each row is base area plus four triangular faces: Pyramid X gives 12 × 12 = 144 in² plus 4 × (½ × 12 × 10) = 240 in², so SA = 384 in². The same plan gives Pyramid Y 64 cm² + 4 × 24 = 160 cm² and Pyramid Z 196 ft² + 4 × 77 = 504 ft².

Turn & TalkSay it out loud with a partner

A square time capsule display has a square and four triangular faces that meet at a point on top. To find its , which parts do you need to measure?

Try starting with:

  • I need the area of the ___ plus the areas of the ___ triangular faces.Necesito el área de la ___ más las áreas de las ___ caras triangulares.
  • The surface area is the ___ area plus the ___ faces.El área de superficie es el área de la ___ más las caras ___.
Use these wordspyramidbaselateral faceslant heightsurface area