Find Areas of Quadrilaterals and Polygons

In this lesson:

• Turn a parallelogram into a rectangle
• Derive the trapezoid formula two ways
• Decompose any polygon into known shapes
• Apply area to real-world coverage problems

What You Will Be Able to Do

By the end, you will:

1. Apply to any parallelogram
2. Derive the trapezoid formula two ways
3. Use on any trapezoid
4. Decompose an irregular polygon into known shapes
5. Use area to solve a real-world coverage problem

A New Shape: The Parallelogram

A parallelogram has two pairs of parallel sides — but no right angles in general.

The slant makes "what is the height?" a real question.

We need a method that turns the unfamiliar into the familiar.

Slide a Triangle to Make a Rectangle

Cut the corner triangle off the left end and slide it to the right. A 7×4 rectangle appears. .

The Area Formula for a Parallelogram

For any parallelogram with base and perpendicular height :

Worked example: ,

Watch Out: The Slant Is Not the Height

A parallelogram has base , slanted side , perpendicular height .

• (a) ← uses the slant
• (b) ← uses the perpendicular height

Off by 7: the size of the missing triangle.

Check-In: Parallelogram with Fractional Height

A parallelogram has base and perpendicular height .

Find the area.

Pause and write your answer.

A Trapezoid Has Two Parallel Bases

A trapezoid has exactly one pair of parallel sides, called and .

The height is the perpendicular distance between them.

Two parallel sides, not one — both must appear in the formula.

Decompose: Cut Into Two Triangles

Draw one diagonal across the trapezoid. It splits into two triangles, both with height .

Two triangles → one formula. The triangle area you already know does the work.

Compose: Two Trapezoids Make a Parallelogram

• Two trapezoids form a parallelogram with base and height

The Area Formula for a Trapezoid

For any trapezoid with parallel bases and and perpendicular height :

The "average of the bases" times the height — both bases matter.

Worked Example: Trapezoid With Bases 6 and 10

, , :

Add the bases first, then multiply by , then halve.

Watch Out: Both Bases Matter

For a trapezoid with , , :

• (a) ← used only
• (b) ← used only
• (c) ← correct

Treating the trapezoid like a triangle drops half the figure.

From Named Shapes to Any Polygon

Triangles, parallelograms, and trapezoids — all derived from the rectangle.

For other polygons (L-shapes, pentagons, and hexagons), there are no new formulas.

Just two strategies: decompose into known shapes, or compose into a bounding rectangle and subtract.

An L-Shape: Two Strategies, Same Answer

A Pentagon as Rectangle Plus Triangle

A "house" pentagon: rectangle base , triangular roof base , altitude .

Decompose the pentagon: rectangle below, triangle above. Sum the two known areas.

Real-World Setup: A Garden Bed

The bed has two parts:

• Trapezoid: feet, feet, feet
• Rectangle extension: feet by feet

One bag of mulch covers square feet. How many bags?

Solving the Garden Bed: Area First

Trapezoid:

Rectangle:

Total: square feet.

From Area to Bags of Mulch

You can't buy of a bag — round up to bags.

The answer to the real question is , not and not .

Watch Out: Match the Units First

A wall is feet wide and inches tall.

Wrong: (mixed units)

✓ Right: inches feet, so square feet

Convert all lengths to one unit first.

The Meta-Move: Find the Right Cut

Every polygon area problem reduces to:

• Identify rectangles, triangles, parallelograms, and trapezoids
• Choose decompose (split) or compose (enclose and subtract)
• Sum or subtract — there are no new formulas after today

Where This Leads: Surface Area

Every face of a 3D figure is a polygon.

In Standard 6.G.A.4, you will:

• Unfold a 3D shape into a flat net
• Find each face's area with today's tools
• Sum face areas for the surface area