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Area of Polygons
& Composites

Learn to calculate area for triangles, quadrilaterals, and composite figures by decomposing complex shapes into simpler ones.

Pre-Algebra

What Is Area?

Area measures the amount of two-dimensional space inside a flat figure. We express area in square units: square centimeters (\(\text{cm}^2\)), square meters (\(\text{m}^2\)), square feet (\(\text{ft}^2\)), and so on.

\(16\) square units
?
Irregular area

The Building-Block Shapes

Every composite figure can be broken into a handful of fundamental shapes. Master these formulas first:

Rectangle
\(A = b \times h\)
Triangle
\(A = \tfrac{1}{2}\,b\,h\)
Parallelogram
\(A = b \times h\)
Trapezoid
\(A = \tfrac{1}{2}(b_1 + b_2)\,h\)

Why Area Matters

Area is not just a classroom exercise: it appears everywhere in real life. Understanding area helps you solve practical problems involving space, materials, and design.

Flooring
How many tiles cover an L-shaped room?
Painting
How much paint for a wall with a triangular gable?
Landscaping
How many bags of seed for a composite yard?

Rectangle & Parallelogram

\(b\)
\(h\)
Rectangle
\(A = b \times h\)
\(b\)
\(h\)
Parallelogram
\(A = b \times h\)

Area of a Triangle

\(h\)
\(b\)
half
A triangle is
half a rectangle
\(A = \tfrac{1}{2}\,b\,h\)

Area of a Trapezoid

\(b_1\)
\(b_2\)
\(h\)

Area vs Perimeter

Area Perimeter
Measures Space inside Distance around
Units \(\text{units}^2\) \(\text{units}\)
Rectangle \(b \times h\) \(2b + 2h\)
Real-world use Carpet, paint, sod Fencing, trim, border

What Is a Composite Figure?

A composite figure is a shape made by combining two or more basic polygons. To find its total area, you decompose (break apart) or compose (add pieces).

A
B
A
\(90 \times 200\)
B
\(70 \times 90\)
+

Step-by-Step Method

1
Identify the basic shapes. Look for rectangles, triangles, parallelograms, or trapezoids hidden inside the figure.
2
Label dimensions. Mark the base and height for each sub-shape. Derive missing lengths from the overall dimensions.
3
Apply the correct formula. Use \(A = b \times h\), \(A = \tfrac{1}{2}\,b\,h\), or \(A = \tfrac{1}{2}(b_1+b_2)\,h\) for each piece.
4
Add or subtract. Sum the sub-areas if the composite is built by joining. Subtract if a region is cut out.
5
Write units. Always include square units (\(\text{cm}^2\), \(\text{ft}^2\), etc.) in your final answer.

Example 1: House Shape

\(10\) m
\(7\) m
\(5\) m
Rectangle
Triangle

Example 2: Trapezoid Garden

\(8\) ft
\(14\) ft
\(6\) ft

Example 3: Subtract to Find Area

Sometimes it is easier to start with a larger shape and subtract the piece that is cut away.

cut
\(12\) cm
\(9\) cm
\(6\) cm
\(4.5\) cm
Rectangle
\(12 \times 9\)
\(-\)
Triangle
\(\tfrac{1}{2}(6)(4.5)\)

Example 4: L-Shaped Room

A
B
\(12\) m
\(5\) m
\(10\) m
\(5\) m
\(5\) m
Part A (rectangle)
\(5 \times 12 = 60\)
Part B (rectangle)
\(5 \times 5 = 25\)
\(60 + 25 = 85\;\text{m}^2\)

Common Mistakes

✕ Using the Slant Side
The height must be perpendicular to the base. Using a slanted side gives the wrong answer for triangles, parallelograms, and trapezoids.
✕ Forgetting the \(\tfrac{1}{2}\)
Triangle and trapezoid formulas both include \(\tfrac{1}{2}\). Skipping it doubles the correct answer.
✕ Wrong Units
Area is in square units (\(\text{cm}^2\)), not linear units (\(\text{cm}\)). Mixing perimeter and area units is a common error.

Formula Summary

Shape Formula Key Note
Rectangle \(A = b \times h\) Sides are perpendicular
Parallelogram \(A = b \times h\) \(h\) is perpendicular, not the side
Triangle \(A = \tfrac{1}{2}\,b\,h\) Half of a rectangle
Trapezoid \(A = \tfrac{1}{2}(b_1+b_2)\,h\) Average the two bases
Composite Sum or difference Decompose, then add/subtract

Question 1 of 5

A parallelogram has a base of 12 cm and a height of 8 cm. What is its area?

Question 2 of 5

A triangle has a base of 14 in and a height of 9 in. What is its area?

Question 3 of 5

A trapezoid has bases of 6 m and 10 m, and a height of 4 m. What is its area?

Question 4 of 5

A composite figure is made of a rectangle (8 ft by 5 ft) topped by a triangle with the same 8 ft base and a height of 3 ft. What is the total area?

Question 5 of 5

A rectangular yard measures 20 m by 15 m. A triangular flower bed with a base of 10 m and height of 6 m is cut from one corner. What is the remaining area?