One standard approach to finding the area of a shape is to divide the shape into subshapes, determine the area of each subshape, and then add the areas together. You have used this approach to answer Problems B1 and B2.
A second approach for finding area is to surround the shape in question with another shape, such as a rectangle. For this approach, you first determine the areas of both the rectangle and the pieces of the rectangle that are outside the original shape, and then you subtract those areas to determine the area of the original shape.
Here are three examples of how to surround a right triangle with a rectangle:
You can also divide a triangle into right triangles, form rectangles around each triangle, and then calculate the areas of the rectangles:
In each case, the area of the triangle is half the area of the rectangle that surrounds it.
Use the rectangle method to find the area of each figure:
Does this method work for non-right triangles? For example, how might you find the area of a triangle like BDE below?
Here's how to do it: First, form rectangle ABCD around BDE. Determine the area of rectangle ABCD and then subtract the areas of ABE and BCD. (Use the rectangle method to determine the areas of these two triangles.) This will give you the area of BDE:
Area of ABCD = 9 square units
Area of ABE = 3 square units
Area of BCD = 4.5 square units
Area of BDE = ABCD - ABE - BCD = 9 - 3 - 4.5 = 1.5 square units
Use the Interactive Activity to work on the geoboard problems in Part B. For a non-interactive version, use an actual geoboard and rubber bands, or print the dot paper worksheet (PDF).
Use this method to find the area of each of the following:
Completely surround the figure with a rectangle; the vertices of the figure should touch the sides of the rectangle. Then find the areas of the outside spaces -- the parts of the rectangle that are not inside the figure in question. Some people like to cover all of the rectangle except the section they are working on so as not to be distracted by overlapping lines and shapes. Close Tip
Video Segment In this segment, Rosalie demonstrates how to use the rectangle method to find the area of a triangle. Watch this segment after you've completed Problems B3 and B4.
For what kinds of figures on the geoboard might this method be particularly useful?
If you are using a VCR, you can find this segment on the session video approximately 8 minutes and 22 seconds after the Annenberg Media logo.
A triangle and a square with equal areas (which one has the smaller perimeter?)
Triangles with areas of 5, 6, and 7 square units, respectively
Use a guess-and-check strategy: First make a triangle on the geoboard. Next determine its area, using one of the methods mentioned earlier. Adjust the shape of your triangle as needed (i.e., make it larger or smaller), and repeat the process, refining the size of your triangle as you get closer to the desired area. Close Tip
Video Segment In this segment, Professor Chapin and Neuza explore what happens to the area of a triangle when its shape is changed, though the height and base lengths remain the same.
Did you come up with a similar conjecture? Explain in your own words why you think this happens.
If you are using a VCR, you can find this segment on the session video approximately 9 minutes and 49 seconds after the Annenberg Media logo.