A Geometric Investigation of (a + b)^2

• ## A Geometric Investigation of (a + b)2

What is the value of (a + b)2? You might think it's a2 + b2, but it's not! Use this geometric demonstration to find out what it really is.

### Instructions

• Adjust the slider to change the lengths of a and b. The shapes below the square will change size accordingly.
• Try to arrange the pieces so that they fully cover the square. Each piece can be moved by dragging the red dot in its upper left corner. Use the Fit Shapes to Square button if you need help arranging the shapes.
• Before pressing the Show Dimensions of Square button, try to figure out what's happening on your own. If you need help, though, press this button.
• To start over, click the Reset button.

### Exploration

Explore this interactive math tool and try to figure out what's happening on your own. How does this demonstration help you discover the value of (a + b)2?

If you need some help, follow the steps below and click on the buttons only when instructed.

1. Arrange the colored shapes so that they completely cover the large white square. When finished, click the Fit Shapes to Square button. This will align the shapes within the square, and it may even show you an arrangement of the pieces that differs from yours.
2. Compare the length of the large white square to the lengths of the smaller shapes. (You can do this by dragging the shapes to the slider at top.)
3. How do the dimensions of the shapes and the large white square compare to the lengths of a and b?
• Click the Show Dimensions of Square button.
• What do you notice about the length of one side of the large white square, in comparison to the lengths of the smaller shapes?
• Move the shapes around to see if this is always true.
• Click the Reset button. Now click the Show Square Dimensions button. Does the information shown confirm your answers to the questions above?

4. What happens if the dimensions change?
• Drag the red dot in the middle of the slider to change the size of a and b.
• Fit the newly sized shapes in the large square by clicking the Fit Shapes to Square button.

5. Determine the area of each of the colored shapes.
• Click the Reset button. Then click the Show Dimensions of Square button. Press the Fit Shapes to Square button to align the shapes.
• What is the area of the large white square?
• What is the area of the yellow square?
• What is the area of the blue square?
• What is the area of the red and green rectangles? Are they equal?

6. Compare the areas.
• What is the sum of the area of the colored shapes?
• Figure out the answer, and then click the Show All Areas button.
• How does the area of the large white square compare to the sum of the areas of the colored shapes?