## Area Formulas: When Students Ask

• Why do I need to know these formulas, and when should I use them?
If students understand how the formulas are derived, it will help them remember them. However, remembering the formulas is not the crucial aspect of formulas. Knowing what formula to use, when to use it, and how to use it correctly to find the solution to a problem are the crucial factors. Formulas can save a lot of time and effort — that is why formulas are so prevalent in mathematics and science.
• Why can't I just multiply the lengths of the two sides to find the area of a parallelogram?
Take four strips of cardboard and tape them together to make a flexible parallelogram. By making the angle between two sides very small, students will see that the area inside the parallelogram will be much less than when the sides form right angles. See the diagram below. This will help students realize that it is not the length of two sides that determines the area, but the length of one side and the perpendicular distance between parallel sides. Let them discuss this phenomenon in class to help them understand the process.
• Does it matter which side is the length and which side is the width of the rectangle when using the formula for the area of a rectangle?
Usually students only see rectangles with the long side parallel to the bottom of the paper. They hardly ever see rectangles in which the long side is vertical and the short side lies horizontally. Similarly, they rarely see rectangles in which no side is parallel to the bottom of the page. Students may think that a rectangle must have the long side parallel to the bottom of the page. They should realize that it doesn't matter which side is called the length and which side is called the width, since the formula for finding the area of a rectangle uses multiplication, and multiplication is commutative (that is, changing the order of the factors does not change the product).
• Does it matter which side is the base of a parallelogram when using the formula for the area of a parallelogram?
It doesn't matter which side is called the base of a parallelogram. Any side can be used, but what is critical is the height of the parallelogram. Students must remember the height of a parallelogram is the perpendicular distance between the sides.