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Let’s further examine the concept of area, using a geoboard. The unit of area on the geoboard is the smallest square that can be made by connecting four nails:
We will refer to this unit as 1 square unit.
On the geoboard, the unit of length is the vertical or horizontal distance between two nails. Perimeter is the distance around the outside of a shape and is measured with a unit of length.
Use the following 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 a space enclosed by five dots both vertically and horizontally to represent a single geoboard.
Problem B1
Make the following figures and find the number of square units in the area of each: Note 2
equals 1 square unit. equals 0.5 a square unit.
Make the following figures:
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 Δ BDE below?
Here’s how to do it: First, form rectangle ABCD around Δ BDE. Determine the area of rectangle ABCD and then subtract the areas of Δ ABE and Δ BCD. (Use the rectangle method to determine the areas of these two triangles.) This will give you the area of Δ BDE:
Area of ABCD = 9 square units
Area of Δ ABE = 3 square units
Area of Δ BCD = 4.5 square units
Area of Δ 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.
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? You can find this segment on the session video approximately 8 minutes and 22 seconds after the Annenberg Media logo. |
Construct the following shapes:
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.
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. You can find this segment on the session video approximately 9 minutes and 49 seconds after the Annenberg Media logo. |
Most of us use formulas to determine the area of common polygons, such as triangles and rectangles. The formula for the area of a rectangle is A = l • w, where l represents the length of the rectangle and w represents the width.
The formula, for the area of a triangle is b•h where b represents the length of the base of the triangle and h represents the height of the triangle. (Height is the length of the segment from a vertex perpendicular to the opposite side.)
Explain how the formulas below relate to using the geoboard to find an area:
Think about the rectangle method you used on the geoboard to find the area of a right triangle.
Why does the formula A = work for triangles other than right triangles? To answer this question, let’s look at parallelograms, since we’ll derive the triangle formula from the formula for the area of a parallelogram.
Problem B7
Note the height of the triangle, h, is the length of the line segment perpendicular to the base and adjoining it to the opposite vertex. This is equal to the height of the parallelogram.
Problem B8
Note 2
If you are working in a group, take this opportunity to learn from one another by sharing your approaches and solution strategies to Problems B1-B4.
Problem B1
Problem B2
Possible solutions:
Problem B3
Problem B4
Problem B5
Problem B6
Problem B7
Problem B8