Composite Area and Volume of Prisms

Learning objectives:

• Compute areas of polygons, including composite 2D figures
• Compute volumes of right prisms with any polygonal base
• Solve real-world problems requiring area or volume
• Select the appropriate measure for a given context

What You Already Know About Prisms

Recall:

• = area of the base
• Multiply by to stretch the base through space
• Today: what if the base isn't a rectangle?

The Decomposition Strategy for Composite Figures

The strategy:

1. Identify the simpler shapes inside the figure
2. Compute each sub-area using known formulas
3. Add or subtract to get the total area

Composite shapes = simpler shapes combined (or cut apart)

Example 1: L-Shaped Floor Plan

Overall: 10 m × 8 m; corner removed: 4 m × 3 m

Example 1: Decompose and Calculate

Approach: Subtraction

• Large rectangle: m²
• Removed corner: m²

Alternative: Split into two rectangles and add — same answer.

Example 2: Field with Semicircular Ends

Rectangle 50 m × 30 m plus two semicircular ends (diameter = 30 m)

Example 2: Decompose and Calculate

Rectangle + two semicircles = rectangle + one full circle:

Check In: Addition or Subtraction?

For each figure, decide: add or subtract?

Practice 1: Wall Panel Problem

A decorative wall panel is 3 m wide and 2 m tall. A circular window of diameter 0.8 m is cut from the center.

What is the painted area of the panel?

Hint: The circle is removed from the panel — which approach?

Practice 1: Wall Panel Answer Revealed

Panel area: m²

Circle area (radius = 0.4 m):

Painted area:

Practice 2: Composite Shape Problem

A composite figure consists of a rectangle 8 cm × 5 cm with a right triangle (base 8 cm, height 3 cm) attached to one end.

Find the total area of the composite figure.

Identify the two shapes, compute each area, then add.

Practice 2: Composite Shape Answer Revealed

Rectangle: cm²

Triangle:

Total area:

Moving From 2D to 3D Figures

Composite 2D area: Break into simpler shapes, add/subtract ✓

Next: Volume of 3D prisms

The key insight: The rectangular prism formula is a special case of a more general rule — one that works for any right prism.

Volume of Any Right Prism:

For any right prism:

• = area of the base (any polygon)
• = height of the prism (distance between the two bases)

For rectangular prisms: , so — the formula you already know.

Base Height vs. Prism Height: A Critical Distinction

Base height: altitude of the triangular cross-section
Prism height: distance between the two triangular faces

Example 1: Rectangular Prism Confirms Formula

Dimensions: 5 cm × 4 cm × 3 cm

Step 1: Find base area

Step 2: Multiply by prism height

Same as — this confirms the general formula works.

Example 2: Triangular Prism Volume Step by Step

in, base height in, prism height in

Base height (6 in) ≠ prism height (8 in).

Check In: What Do You Need First?

To compute , you need:

• Base area — 2D formula for the base shape
• Prism height — distance between the two bases

Find : trapezoidal base, parallel sides 5 ft and 9 ft, trapezoid height 4 ft.

Example 3: Trapezoidal Prism Volume Step by Step

Trapezoidal base: parallel sides 5 ft and 9 ft, trapezoid height 4 ft; prism height 7 ft

Step 1: Base area (trapezoid formula)

Step 2: Volume

Practice 3: Triangular Prism Volume

A triangular prism has a right-triangle base with legs 9 cm and 12 cm, and a prism height of 15 cm.

Find the volume. Show both steps.

Step 1: Find B. Step 2: Multiply by h.

Practice 3: Triangular Prism Answer Revealed

Step 1: Base area (right triangle)

Step 2: Volume

For a right triangle, the two legs are the base and height — no separate altitude needed.

Deck 1 Summary: Key Takeaways

What Comes Next: Deck 2

Coming up:

• Surface area: nets, all faces, for prisms
• Slant height vs. vertical height for pyramids
• Choosing the right measure: area, surface area, or volume?

Prerequisite: Composite area and from today.