01 / 20

Find Area by
Composing & Decomposing

Learn to find the area of triangles, special quadrilaterals, and polygons by breaking them into simpler shapes or building rectangles around them.

6th Grade Math

What Is Area?

Area measures the amount of flat space a shape covers. We measure area in square units such as \(\text{cm}^2\), \(\text{in}^2\), or \(\text{ft}^2\).

\(1\text{ unit}^2\)
 5 units 4 units
\(= 20\text{ unit}^2\)

Two Big Strategies

Decompose
Break a complex shape apart into simpler pieces (triangles, rectangles). Find each area, then add them together.
Compose
Build a larger rectangle around the shape. Find the rectangle's area, then subtract the extra parts.

Decompose vs Compose: A Visual

Decompose A B + A B Compose extra extra Rectangle − extras = Triangle

Area of a Triangle

b (base) h (height)
\(\text{Area} =\)
\(\frac{1}{2} \times b \times h\)
= half the rectangle

Area of a Parallelogram

b (base) h
\(\text{Area} =\)
\(b \times h\)

Area of a Trapezoid

\(b_1\)
\(b_2\)
 h
\(\text{Area} =\)
\(\frac{1}{2}(b_1+b_2) \times h\)

When to Decompose vs Compose

Decompose Compose
What you do Cut shape into pieces Surround shape with a rectangle
Then Add the piece areas Subtract the extra areas
Best for L-shapes, plus signs, concave figures Triangles, irregular polygons
Key skill Spot clean cut lines Identify leftover triangles

How to Apply the Strategy

1
Study the shape. Identify what kind of polygon it is and look for right angles or parallel sides.
2
Choose your strategy: Decompose (cut into pieces) or Compose (surround with a rectangle).
3
Draw your cuts or your surrounding rectangle. Label every base and height you can find.
4
Calculate each piece's area using rectangle or triangle formulas.
5
Combine: Add (if you decomposed) or subtract (if you composed) to get the total area.

Example 1: Right Triangle

12 cm 9 cm A B C
\(\text{Area} =\)
\(\frac{1}{2} \times 12 \times 9\)
\(= 54\text{ cm}^2\)

Example 2: L-Shaped Room

A B 11 ft 5 ft 6 ft 6.5 ft
Decompose:
\(A = 11 \times 5 = 55\text{ ft}^2\)
\(B = 6 \times 6.5 = 39\text{ ft}^2\)
\(\text{Total} = 94\text{ ft}^2\)

Example 3: Irregular Pentagon

\(T_1\)
\(T_2\)
 14 m 12 m
\(b=8.5, h=5\)
\(b=5.5, h=3.5\)
Compose:
\(\text{Rect} = 14 \times 12 = 168\)
\(T_1 = \frac{1}{2}(8.5)(5) = 21.25\)
\(T_2 = \frac{1}{2}(5.5)(3.5) = 9.625\)
\(168 - 30.875\)
\(= 137.125\text{ m}^2\)

Example 4: Trapezoid Area

8 in 16 in 7 in
\(\text{Area} =\)
\(\frac{1}{2}(8 + 16) \times 7\)
\(= \frac{1}{2} \times 24 \times 7\)
\(= 84\text{ in}^2\)

Real-World Application

A homeowner wants to tile an oddly shaped patio. The patio is a rectangle with a triangular bump-out on one side.

12 ft 7.5 ft 6 ft 7.5 Rect Tri

Common Mistakes

✕ Using the slant side as height
The height must be perpendicular to the base. A slanted side is longer than the true height and gives the wrong answer.
✕ Forgetting the ½ in triangles
A triangle is half a rectangle. If you compute \(b \times h\) without dividing by 2, your area is double the correct value.
✕ Adding when you should subtract
When composing (surrounding with a rectangle), you must subtract the extra regions, not add them.

Formula Summary

Shape Formula Key Detail
Rectangle \(l \times w\) Base case for all others
Triangle \(\frac{1}{2} b h\) Height ⊥ to base
Parallelogram \(b \times h\) Not the slant side
Trapezoid \(\frac{1}{2}(b_1 + b_2) h\) Average the two bases
Any polygon Compose or Decompose Add or subtract pieces

Question 1 of 5

A triangle has a base of 10 cm and a height of 6 cm. What is its area?

Question 2 of 5

A parallelogram has a base of 9 in and a height of 4 in. Its slant side is 5 in. What is the area?

Question 3 of 5

A trapezoid has bases of 5 m and 11 m, and a height of 8 m. What is its area?

Question 4 of 5

An L-shaped figure is made of two rectangles: one is 10 ft × 4 ft and the other is 3 ft × 6 ft. What is the total area?

Question 5 of 5

A student draws a rectangle around an irregular polygon. The rectangle is 20 cm × 15 cm. The extra corner triangles have areas of 30 cm², 22.5 cm², and 15 cm². What is the polygon's area?