Area of Triangles, Quadrilaterals, and Polygons

Lesson 1 of 1: 6.G.A.1

In this lesson:

• Find area of all triangle types using
• Find area of parallelograms and trapezoids
• Decompose composite polygons into simpler shapes

Learning Objectives for This Lesson

By the end of this lesson, you will:

1. Apply for any triangle — right, acute, or obtuse
2. Apply for parallelograms and for trapezoids
3. Find the area of composite polygons by decomposing into simpler shapes

What You Already Know About Area

You already know rectangle area:

• A 4 × 6 rectangle: square units
• Area counts square units inside a shape
• Every square is a special rectangle:

Rectangles are the key to every other shape.

Every Triangle Is Half a Rectangle

Every triangle fits exactly half of an enclosing rectangle.

Deriving the Triangle Area Formula

The enclosing rectangle has area .

The triangle is half of that rectangle:

• Base (): any side of the triangle
• Height (): perpendicular distance from opposite vertex to the base

The Height Can Be Outside the Triangle

• Right: height is one of the legs
• Acute: height is inside the triangle
• Obtuse: height falls outside — draw as a dashed line to the extended base

Worked Example: Right Triangle Area

Given: legs of 6 cm and 8 cm

Step 1: Identify base and height — the legs are perpendicular

Step 2: Apply the formula

Acute and Obtuse Triangle Examples

Acute — base = 10 m, height = 5 m (inside):

Obtuse — base = 12 ft, height = 4 ft (outside):

Height outside the triangle? Use it anyway — same formula.

Check-In: Which Measurement Is the Height?

Triangle A: sides 5, 8, 10; dashed perpendicular line of 4 drawn outside the triangle

Triangle B: right triangle, legs 7 and 9, slant side 11

Identify the height in each triangle. Find the area.

Check-In: Answers to Both Triangles

Triangle A: height = 4 (the dashed perpendicular outside)

Triangle B: height = 7 (using base = 9)

The slant side of 11 is never the height.

Connecting Triangles to Parallelograms and Trapezoids

We proved every triangle = half a rectangle.

Now: can we do the same for parallelograms?

• A parallelogram looks like a "slanted rectangle"
• Same base, same height — same area?

Yes — and the method is called "cut and slide."

Parallelogram Area via Cut and Slide

• Cut a triangle from one end
• Slide it to the other end → rectangle
• Same base, same height:

The slant side is NOT the height.

Worked Example: Parallelogram Height vs Slant

Given: base = 8 cm, slant side = 5 cm, height = 4 cm

Which measurement is the height?

Not — the slant side is a side length, not a height.

Trapezoid Formula: Averaging Two Parallel Bases

A trapezoid has two parallel sides (the bases): and

Deriving the formula:

• Average the two bases:
• Multiply by the height

This equals the area of a rectangle with base and height .

Worked Example: Finding Trapezoid Area

Given: m, m, m

Step 1: Add the two bases

Step 2: Apply the formula

Quick Check: Parallelogram or Trapezoid?

Find the area of each shape:

Shape P: parallelogram — base 6 ft, slant side 7 ft, height 5 ft

Shape T: trapezoid — in, in, height = 6 in

Compute both areas before advancing.

Quick Check: Answers for Both Shapes

Shape P (parallelogram):

(Not — the slant side is not the height)

Shape T (trapezoid):

Strategy: Decompose Composite Polygons Into Pieces

Rule: Draw the decomposition lines before computing anything.

Worked Example: L-Shape Area by Decomposition

• Top rectangle: 4 × 3 →
• Bottom rectangle: 8 × 5 →

Multiple decompositions are valid — the total is always the same.

The Enclose and Subtract Area Method

Sometimes it's easier to enclose, then subtract.

Method:

1. Draw the smallest rectangle that encloses the shape
2. Find the enclosing rectangle's area
3. Subtract the areas of pieces that don't belong

Example: Rectangle with Triangular Notch

Setup: A 10 × 8 rectangle with a right-triangle notch cut from one corner (legs 3 and 4)

Bounding rectangle:

Triangular cutout:

Real-World Application: T-Shaped Garden Area

A T-shaped garden: outer width 10 m, top section 3 m tall, lower section 4 m wide × 5 m tall

Step 1: Split into two rectangles
Step 2: Find each area and add

Try before advancing.

Garden Floor Plan Solution: Decomposition Method

Top rectangle: 10 × 3 →

Bottom rectangle: 4 × 5 →

Alternative: bounding 10 × 8 = 80 minus corner cutouts.

Check-In: Choose the Easier Area Method

A hexagonal patio has an outer rectangle 12 ft × 8 ft, with right-triangle corners cut off. Each corner triangle has legs 2 ft and 3 ft.

Which method is easier — decompose or compose-and-subtract?

Find the patio area before advancing.

Check-In: Hexagonal Patio Area Solution

Bounding rectangle:

Four corner triangles:

Composition-subtraction is simpler here — fewer pieces to track.

Key Takeaways and Misconception Warnings

✓ for every triangle — right, acute, or obtuse

✓ parallelograms; trapezoids

✓ Composite polygons: draw decomposition lines first, then compute

Slant side is never the height — find the perpendicular

Triangles: ; parallelograms: — don't confuse them

L-shape: draw the split line first — don't use the bounding rectangle

What Comes Next: Volume and 3D Shapes

Next lesson: 6.G.A.2 — Volume of Rectangular Prisms

• Volume = base area × height
• The base area comes from everything you learned today
• Triangular prisms, rectangular prisms, and more

Today's area formulas are the foundation for 3D geometry.