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Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Triangular Prisms, Pyramids, and Real-World SA

In this lesson:

  • Surface area of a triangular prism
  • Surface area of a square pyramid and tetrahedron
  • Real-world problems: gift wrap, tents, painted rooms
Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

What You Will Be Able to Do

By the end, you will:

  1. Compute surface area of a triangular prism (2 triangles + 3 rectangles)
  2. Compute surface area of a square pyramid and a tetrahedron, using slant height
  3. Solve real-world surface area problems — including problems with selected faces
Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

How Much Canvas Covers a Tent?

A tent is a triangular prism lying on its side.

  • Two sloped sides — canvas
  • Floor — canvas
  • Two triangle ends — open mesh, no canvas

Not every face counts.

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

From Rectangular to Triangular: Morph the Net

Take a rectangular prism's net.

Replace one pair of opposite rectangles with triangles.

The figure becomes a triangular prism:

  • 2 triangles (the "ends")
  • 3 rectangles (the "walls")
Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Triangular Prism Net: 2 Triangles + 3 Rectangles

A 3D triangular prism on the left with right-triangle base (legs 3 and 4, hypotenuse 5) and prism length 6. On the right, the flat net: 2 right triangles at top and bottom, and 3 rectangles between them with widths 3, 4, and 5 and length 6

The 3 rectangle widths are exactly the 3 sides of the triangle.

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Worked Example: SA of a Right-Triangle Prism

Triangle base: legs 3 and 4, hypotenuse 5. Prism length 6.

  • Triangles (2 faces): each , pair total
  • Rectangles (3 faces): , , , total

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Two Heights, One Word: Be Careful

In a triangular prism, the word "height" can mean two things:

  • Triangle altitude — used for the triangle's area:
  • Prism height (or prism length) — used for the rectangles' length

Always label both on every diagram.

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Prism Formula:

Generalize the worked example:

  • Triangles total:
  • Rectangles total:

For our prism: ✓

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Square Pyramid Net: 1 Square + 4 Triangles

A 3D square pyramid on the left with base side 6 and slant height 5. On the right, the flat net: 1 central square (6 by 6) with 4 congruent triangles fanned out from its sides, each triangle with base 6 and height 5 marked as the slant height

The 4 triangles meet at the apex when folded up.

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Slant Height vs Vertical Height

A cutaway diagram of a square pyramid showing both heights distinctly. A solid line from the base edge midpoint up the face to the apex is labeled "slant height" — used for face area. A dashed line from the base center straight up to the apex is labeled "vertical height" — used for volume in later grades

For surface area, always use the slant height — it's each triangle face's altitude.

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Worked Example: Square Pyramid SA

Base side 6, slant height 5.

  • Square base:
  • Triangles (4): each , total

Formula: .

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Tetrahedron Has 4 Congruent Triangles

A regular triangular pyramid has 4 identical triangular faces.

With base 4 and face altitude 3:

If the faces aren't congruent, sum each triangle's area separately.

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Check-In: Identify the Figure and Count Faces

Each net below has 5 faces. Which is which?

Figure A Figure B
2 triangles + 3 rectangles 1 square + 4 triangles
___ ? ___ ?

Commit before the next slide.

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Guided Practice: Isosceles-Base Triangular Prism

Isosceles base: sides 5, 5, 6; altitude 4. Prism length 8.

  • Triangle (each): $\tfrac{1}{2}(6)(4) = $ ___; pair = ___
  • Rectangles: , , ; total = ___

$SA = $ ___ + ___ = ___ sq units

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Real Problems Decide Which Faces Count

Surface area is a tool, not a fixed number.

  • Wrapped present: all 6 faces
  • Tent canvas: only 3 faces
  • Painted room: walls + ceiling, no floor

The problem decides which faces.

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Three Real-World Contexts Side by Side

Context Figure Faces
Gift wrap Rect prism All 6
Tent canvas Tri prism 3 (walls + floor)
Painted room Rect prism 5 (no floor)

Same procedure; context picks faces.

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Worked Example: Gift Wrap a 12×8×4 Box

A 12 by 8 by 4 inch box drawn in 3D with wrapping paper around it. Net of 6 rectangles labeled with dimensions and partial pair products: 12x8 = 96, 12x4 = 48, 8x4 = 32

Convert: sq ft → buy 3 sq ft

⚠️ Use ÷ 144, not ÷ 12 ( sq ft sq in)

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Worked Example: Tent Canvas — Selected Faces

A triangular-prism tent. Triangle base: 6 ft base, 4 ft equal sides, 3 ft altitude. Prism length 8 ft. The 2 sloped rectangle walls (4 by 8) and 1 floor rectangle (6 by 8) are highlighted as canvas. The 2 triangular ends are shown as open mesh

  • 2 sloped walls: each, total
  • Floor:
  • Triangle ends: open mesh, not canvas

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

What If: Painting a Room

Room 12 ft × 10 ft × 8 ft. Paint walls and ceiling, not floor.

  • 2 walls: each → 192
  • 2 walls: each → 160
  • Ceiling:

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Check-In: One Face of a Triangular Prism

Triangle base: legs 5 and 12, hypotenuse 13. Prism length 4.

What is the area of one rectangle face of dimensions ?

Commit alone first.

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Your Turn: Cardboard Cost for Packaging

A cardboard box for a in product at $0.005 per sq in.

  1. Identify figure; decide faces (closed → all 6)
  2. Compute surface area
  3. Multiply by cost per sq in
Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

Common Errors to Watch For

⚠️ Using vertical height instead of slant height (pyramid)

⚠️ Forgetting the base of a pyramid

⚠️ Swapping prism height and triangle altitude

⚠️ Converting sq in → sq ft by ÷ 12 instead of ÷ 144

⚠️ Computing full SA when faces are excluded (or vice versa)

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

The Meta-Move for Every Surface Area Problem

For any real or mathematical problem:

  1. Identify the 3D figure
  2. Sketch or visualize the net
  3. Decide which faces are included
  4. Sum the relevant face areas
  5. Apply units and any downstream computation

Same five steps every time. The procedure is universal.

Grade 6 Mathematics | Standard 6.G.A.4
Triangular Prisms, Pyramids, and Real-World SA | Lesson 2 of 2

What Comes Next in Grades 7 and 8

Grade 7 (7.G.B.6):

  • Composite figures — L-shapes, house with roof

Grade 8 (8.G.C):

  • Cylinders, cones, spheres — curved surfaces

The net-and-sum logic stays the same.

Grade 6 Mathematics | Standard 6.G.A.4