Lesson 1: Reasoning to Find Area

Property

To find the area of a general polygonal figure, you can partition it into a combination of simpler shapes like rectangles and triangles, for which area formulas are known. The area of the original figure is the sum of the areas of the non-overlapping component figures.

Examples

• An L-shaped desk can be split into two rectangles: one $4 \times 2$ ft and another $5 \times 2$ ft. The total area is $(4 \times 2) + (5 \times 2) = 8 + 10 = 18$ square feet.
• A shape is made of a $6 \times 6$ cm square with a triangle on top. The triangle's base is 6 cm and height is 4 cm. The total area is $(6 \times 6) + (\frac{1}{2} \times 6 \times 4) = 36 + 12 = 48$ square cm.
• A polygon is composed of a central $10 \times 5$ rectangle and two identical triangles on each side, each with a base of 3 and height of 5. The area is $(10 \times 5) + 2 \times (\frac{1}{2} \times 3 \times 5) = 50 + 15 = 65$ square units.

Explanation

Don't have a formula for a weird shape? Just chop it up! By breaking a complex polygon into familiar pieces like rectangles and triangles, you can find the area of each piece and add them all up for the total.

Property

To find the area of a composite shape formed by removing a smaller rectangle from a larger one, subtract the area of the smaller rectangle from the area of the larger rectangle.

Examples

Property

The area of a shape remains the same when it is cut into pieces and rearranged into a new shape without any gaps or overlaps. This method transforms a complex shape into a simpler one, like a rectangle or square, to make calculating the area easier.

Examples

• A parallelogram with base $b=8$ and height $h=5$ can be rearranged into a rectangle. By cutting a triangle from one side and moving it to the other, we form a rectangle with dimensions $8 \times 5$. The area is $8 \times 5 = 40$ square units.
• A trapezoid with bases $b_1=6$ and $b_2=10$ and height $h=4$ can be transformed. By making a cut through the midpoints of the non-parallel sides and rotating the top section, a parallelogram is formed with base $b_1+b_2=16$ and height $\frac{h}{2}=2$. Or, more simply, it can be rearranged into a rectangle with length $\frac{b_1+b_2}{2} = 8$ and width $h=4$, giving an area of $8 \times 4 = 32$ square units.

Explanation

This skill involves visually decomposing a shape and then recomposing the pieces into a more familiar figure. It's a powerful reasoning tool that shows area is conserved even when a shape's form changes. Instead of just adding or subtracting areas, you are transforming the shape itself. This method is particularly useful for deriving area formulas for shapes like parallelograms and trapezoids by relating them back to rectangles.