Surface Area, Pyramids, and Choosing Measures

Learning objectives:

• Compute surface areas of right prisms and pyramids by decomposing into faces
• Solve real-world problems requiring area, volume, or surface area
• Select the appropriate measure for a given real-world context

What You Will Learn in Deck Two

By the end of this deck, you will be able to:

1. Compute surface areas of right prisms using
2. Compute surface areas of pyramids using slant height
3. Solve real-world problems requiring area, volume, or surface area
4. Choose the correct measure before computing

What Covers a Box? The Net Approach

How much cardboard does a cereal box use?

The box has 6 faces: top, bottom, front, back, and two sides.

To find the total area, we need every face's area.

Strategy: unfold into a flat net, label each face, then sum.

Net of a Rectangular Prism

All 6 faces visible: 2 bases (top/bottom) + 4 lateral rectangles

The SA Shortcut Formula for Right Prisms

For any right prism:

• = area of one base
• = perimeter of the base
• = prism height

Why: = two bases; = lateral faces unrolled into one rectangle.

Example 1: Rectangular Prism SA

8 cm × 5 cm × 3 cm

Verify: cm² ✓

Example 2: Equilateral Triangle Prism SA

Side = 4 cm, area ≈ 6.93 cm²; prism height = 10 cm

3 lateral faces: each 4 cm × 10 cm.

Check In: Face Count for a Triangular Prism

How many faces does a triangular prism have?

List them:

• Bases: _____ faces
• Lateral faces: _____ rectangles
• Total: _____ faces

Check: in SA = 2B + Ph, what does Ph account for?

Pyramid SA: Slant Height Is the Key

Lateral face area uses slant height — not the vertical height

Example 3: Square Pyramid Surface Area

Base: 6 cm × 6 cm; slant height = 5 cm

Use slant height (5 cm), not the vertical height.

Practice 1: Triangular Prism Surface Area

A triangular prism has an isosceles right triangle base with legs 6 cm each, and a prism height of 8 cm.

(Hypotenuse ≈ 8.49 cm; triangle area = 18 cm²)

Find SA using .

Practice 1: Triangular Prism Answer Revealed

Three lateral faces: two rectangles 6 × 8 and one 8.49 × 8.

Practice 2: Rectangular Pyramid Surface Area

A rectangular pyramid has a 10 cm × 6 cm base. Faces on the 10 cm sides: slant height 7 cm. Faces on the 6 cm sides: slant height 9 cm.

Find the total surface area.

Practice 2: Rectangular Pyramid Answer Revealed

Connecting Surface Area to Real-World Choices

Surface area of prisms and pyramids ✓

In real problems, no one labels which formula to use:

"Water fills the tank?" → Volume
"Glass to build the tank?" → Surface area
"Carpet for the floor?" → Area

Identify what is being measured before computing.

Choosing the Right Measure: Four Options

Worked Example: Fish Tank Needs Two Measures

A fish tank is 60 cm × 30 cm × 35 cm.

(a) Volume of water: cm³ L

(b) Glass for 5 faces (no top):

Worked Example: Tent Canvas Problem

Triangular prism tent: triangular face (base 2 m, height 1.5 m), length 3 m.

Canvas needed: 2 triangular ends + 3 rectangular panels (no floor)

Each rectangular panel is 3 m long — widths come from triangle's sides.

Worked Example: L-Shaped Garden Problem

An L-shaped garden: overall area = 18 m², outer perimeter = 20 m.

• Area to be seeded → 18 m² (2D interior)
• Fencing needed → 20 m (going around the boundary)

Same figure — two different measures. Read the question first.

Check In: Which Measure Do You Need?

For each scenario, choose: area / surface area / volume / perimeter

1. Paint the outside of a cylindrical can?
2. Fill a raised garden bed with soil?
3. Rope to outline a rectangular field?
4. Wrapping paper for a gift box?
5. Tile a bathroom floor?

Practice 3: Two-Part Prism Problem

A rectangular prism: 12 cm × 8 cm × 5 cm.

(a) Find the full surface area.
(b) One 8 × 5 face is left open. What area needs covering?

Use , then subtract the open face.

Practice 3: Two-Part Prism Answer Revealed

(a) cm², cm

(b) Open face = cm²

Deck 2 Summary: Key Warnings

Preview of the Next Lesson

You've completed 7.G.B.6.

Coming next: 8th grade extends these ideas (8.G.C.9)

• Cylinder: — same pattern, circular base
• — same net approach

The prism ideas you learned today transfer directly.