Lesson 4: Find Areas of Polygons

Property

The area of a composite shape can be found by adding the areas of its individual parts or by subtracting areas from a larger, simpler shape.
Addition: $A_{total} = A_1 + A_2$
Subtraction: $A_{total} = A_{larger} - A_{smaller}$

Examples

• A shape is formed by a $5 \times 4$ rectangle and a triangle with a base of $4$ and a height of $3$ attached to one side. The total area is the area of the rectangle plus the area of the triangle: $A = (5 \times 4) + \frac{1}{2}(4 \times 3) = 20 + 6 = 26$.
• A large $10 \times 12$ rectangular garden has a $3 \times 4$ rectangular shed inside it. The area of the garden space is the area of the large rectangle minus the area of the shed: $A = (10 \times 12) - (3 \times 4) = 120 - 12 = 108$.

Explanation

Many complex shapes can be viewed as combinations of simpler shapes like rectangles and triangles. To find the area of such a shape, you can break it down into these basic, non-overlapping components and add their individual areas together. Alternatively, if a shape has a hole or a section removed, you can calculate the area of the larger outer shape and subtract the area of the removed inner shape. This addition/subtraction method is a powerful strategy for finding the area of any polygon.

Session 4. Area on the Coordinate Plane

Property

Once a polygon is graphed on a coordinate plane, you can use the grid to measure its base and height. Then, apply the standard area formulas you already know:

• Rectangle:
  $$A = bh$$
• Parallelogram:
  $$A = bh$$
• Triangle:
  $$A = \frac{1}{2}bh$$
• Trapezoid:
  $$A = \frac{1}{2}h(b_1 + b_2)$$

Examples

• A rectangle has a base of 4 units and height of 7 units. Its area is 4 * 7 = 28 square units.
• A triangle has a horizontal base on the grid that is 9 units long, and its highest point is 9 units straight up from the base. The area is 1/2 * 9 * 9 = 40.5 square units.