The area of a rectangle can be found when we know the lengths of its sides. Is the same true for a parallelogram? That is, can we find the area of a parallelogram when we know the lengths of its sides? Why or why not?
No, we cannot find the area of a parallelogram when we only know the lengths of its sides.
The area of a parallelogram is given by the formula base × height. While the base corresponds to one of the side lengths, the height is the perpendicular distance between the bases, which cannot be determined from the side lengths alone.
A parallelogram can be 'thin' or 'wide' depending on its angles. Even with the same side lengths, different angles will result in different heights, and consequently, different areas. Therefore, knowing the side lengths is insufficient; we must know the height to find the area.
Explanation
The textbook context specifies that the area of a parallelogram is calculated as base × height. It also introduces the concept of a 'thin' parallelogram, which highlights that the shape and internal angles can vary greatly even if the side lengths remain constant. Because the height depends on these angles and not just the side lengths, knowing only the sides does not provide enough information to calculate the area.
Solution Steps
Step 1: Identify the formula for the area of a parallelogram, which is base × height.
Step 2: Note that knowing the lengths of the sides only provides the base, not the height.
Step 3: Conclude that since a parallelogram with the same side lengths can be 'thin' or 'wide' (having different heights), the area cannot be determined without knowing the height.