You should already know that the area of a rectangle is the product of the length of the base and the height, or A = bh. On this web page you'll extend that concept to discover or demonstrate the formula for the area of a parallelogram.

Sketch

This sketch starts with what appears to be one rectangle. Drag the point labeled Parallelogram to skew the top base and make a parallelogram.

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Investigate

1. As you drag to create a parallelogram, carefully look at the shaded areas of the rectangle and parallelogram. How do they compare? Is your visual observation supported by the area measurements?
2. The pink segment represents the height of the parallelogram. How does it compare to the height of the rectangle? Drag the point Adjust Height. Do the heights of the parallelogram and the rectangle still compare in the same way? What happens to the area measurements as you drag Adjust Height?
3. Notice that the base of the parallelogram and the base of the rectangle are the same segment. What happens to the area measurements as you drag the point Adjust Base?
4. From what you've seen so far, how would you calculate the area of a parallelogram given the length of its base and its height?

Sketch

This sketch starts a parallelogram. It is actually made of two pieces; you can move the end piece by dragging point O. You can change the shape and size of the parallelogram by dragging the red points and pressing Close Up Parallelogram to put the pieces back together. If you accidentally drag a point off the screen, press Start Over several times until everything is within reach again.

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Investigate

1. Use point O to break the parallelogram into its two pieces. Try arranging the pieces into other shapes without overlapping them. Is the area of each of these new shapes the same as the area of the original parallelogram? Why?
2. Can you rearrange the pieces to form a rectangle? (Press Arrange Pieces if you need help.) What is the area of the rectangle in terms of b, h, and s? What does this mean about the area of the parallelogram?
3. Use the points and buttons to repeat steps 1–3 for a different parallelogram. Can you make a rectangle out of any parallelogram? Is the area calculated in the same way?
4. Formulate the Parallelogram Area Conjecture: The area of a parallelogram is given by the formula _____, where A is the area, b is the length of the base, and h is the height of the parallelogram.