Neft Teacher · Learn It

Unit 5 · Standard 6.GR.4

Determine Surface Area of Prisms

Lesson 5-7 · Learn It — how the math works

Why we're learning this

Your class is painting the outside of different time capsule shapes to protect them from weather. Some capsules are rectangular prisms and others are triangular prisms (wedge-shaped). To buy the right amount of paint, you need to calculate the surface area of each shape! One capsule is a rectangular prism 7 inches long, 4 inches wide, and 3 inches tall. Another is a triangular prism whose triangle bases are 8 inches wide and 3 inches tall, with slant sides of 5 inches, and the prism is 9 inches long.

✱ Math Workbench

Learn It

How do we find the surface area of a prism?

Surface area is the total area of all the faces of a solid. A rectangular prism has 6 faces; a triangular prism has 5 faces (2 triangles + 3 rectangles). Add every face to get the total.

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

James is putting a restorative stain on the entire outside of the wooden chest. One can of stain covers about 50 square feet.
Where you see this in real lifeJames is putting a restorative stain on the entire outside of the wooden chest. One can of stain covers about 50 square feet.

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

The problem looks like this
One triangle base of the prism: base 6 in., height 4 in., slant sides 5 in. each.6 in.4 in.5 in.5 in.
  1. My has triangle bases (base 6 in., height 4 in.), is 10 in. long, and the two slant sides are 5 in. each.
  2. Each triangle base is ½ × 6 × 4 = 12, and there are 2 of them: 2 × 12 = 24.
  3. Now the 3 rectangles: 6 × 10 = 60, 5 × 10 = 50, and 5 × 10 = 50, which add to 160.6 × 10 = 60, 5 × 10 = 50
  4. I add the bases and the rectangles:24 + 160 = 184
  5. So the is 184 in². I counted 5 because it is a .

Watch out — mistakes students make here

  • 144 is the volume — surface area sums all six faces.
  • Surface area uses square units (ft²), not cubic.
  • Check the units — a wrapped surface is measured in square units, not cubic.

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

Same steps, new numbers.

  1. Now a : 5 long, 4 wide, 3 tall. How many does it have?
  2. Top and bottom: 5 × 4 = 20, times 2 = 40. Front and back: 5 × 3 = 15, times 2 = 30.5 × 4 = 20
  3. Left and right: 4 × 3 = 12, times 2 = 24. Now add 40 + 30 + 24. What is the ?4 × 3 = 12
  4. Yes, SA = 94 cm². always uses square units.

On your ownYou do — solve this one by yourself

Before you start, remember:

  • In the table below, find the total of each capsule, whether it is rectangular or triangular.
  • Add up every and write your answer in square units.

Solve: Calculate the of each time capsule.

CapsuleLengthWidthHeightSurface Area
Capsule E12 in8 in5 in
Capsule F9 cm6 cm6 cm
Capsule G15 ft4 ft3 ft

Stuck? Open one hint at a time.

Hint 1

Each capsule needs SURFACE AREA: the sum of all six face areas, written in square units.

Hint 2

Row by row, use SA = 2lw + 2lh + 2wh — double each of the three different faces, then add.

Hint 3

Capsule E: the faces are 12×8 = 96, 12×5 = 60, and 8×5 = 40. Double each, then add the three results.

Check my answer
✅ Answers:
  • Capsule E → Surface Area: 392 in²
  • Capsule F → Surface Area: 288 cm²
  • Capsule G → Surface Area: 234 ft²
Each capsule is wrapped on all six faces, so use SA = 2lw + 2lh + 2wh. Capsule E gives 2(96) + 2(60) + 2(40) = 392 in², Capsule F gives 2(54) + 2(54) + 2(36) = 288 cm², and Capsule G gives 2(60) + 2(45) + 2(12) = 234 ft² — and each answer keeps the square units of its row.

Turn & TalkSay it out loud with a partner

Your class is painting two capsule shapes: a and a . How is finding the of each shape the same, and how is it different?

Try starting with:

  • Both surface areas are found by adding the areas of all the ___.Ambas áreas de superficie se hallan sumando las áreas de todas las ___.
  • A rectangular prism has ___ faces, but a triangular prism has ___ faces.Un prisma rectangular tiene ___ caras, pero un prisma triangular tiene ___ caras.
Use these wordssurface arearectangular prismtriangular prismfacelateral face