Lesson 2: Area of Parallelograms

Property

A parallelogram can be rearranged into a rectangle with the same base and height. Choose any side of the parallelogram as the base (length $b$), and let $h$ be the perpendicular distance between the base and the opposite side.
The area is given by the formula:

Examples

• A parallelogram has a base of 12 cm and a height of 5 cm. Its area is $12 \times 5 = 60$ square cm.
• A section of a patio is shaped like a parallelogram with a base of 8 feet and a height of 6 feet. The area is $8 \times 6 = 48$ square feet.
• Even if the slanted side is 9 inches, if the base is 15 inches and the height is 7 inches, the area is $15 \times 7 = 105$ square inches.

Explanation

Think of a parallelogram as a slanted rectangle. By slicing off a triangle from one end and moving it to the other, you create a perfect rectangle. This new rectangle has the same base and height, which is why the area formula works!

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.

Property

Parallelograms that share the same base ($b$) and have the same perpendicular height ($h$) have equal areas. This is a direct consequence of the area formula, $A = b \times h$.

Examples