## Join us for conversations that inspire, recognize, and encourage innovation and best practices in the education profession.

**Available on Apple Podcasts, Spotify, Google Podcasts, and more.**

- Measuring a Surface
- Areas of Irregular Shapes

Think of some situations that involve the measurement of area. How would you determine these areas? For example, in order to determine how much paint to buy for your living room, how would you find the area of each surface you plan to paint — your walls, ceiling, etc.?

Units of measurement for area involve shapes that cover a plane. Some shapes, such as rectangles and squares, cover the plane more completely than other shapes, such as circles. As mentioned above, squares are the most common unit of measurement in the U.S. customary system (square inches, square feet, square yards, square miles). The standard unit in the metric system is the square meter, with square centimeters and hectares (equivalent to 10,000 square meters) used for smaller and larger surfaces.

Most people are familiar with methods for finding the area of familiar shapes, such as rectangles and squares. But is it possible to determine the area of an irregularly shaped object? How would you determine such an area without a formula? Let’s explore.

Trace your hand on a piece of paper. Think about how you might determine the area of your handprint.

**Problem A1**

Will the amount of area covered differ if you trace your hand with your fingers close together or spread apart? Explain.

Think about the activity you did with the tangram triangles in Session 1, Part C.

Units for measuring area must have the following properties:

- The unit itself must be the interior of a simple closed shape.
- The unit, when repeated, must completely cover the object of interest, with no holes or gaps (like a tessellation).
- Many polygons (e.g., rectangles, rhombuses, and trapezoids) and irregular shapes (e.g., L shapes) have this property and can thus be used as units of measurement.

**Problem A2**

- What units might you use to determine the area of your handprint?
- Why can’t you use a small circle as the unit of measurement?

**Problem A3 **

- One method for finding the area of an irregular shape is to count unit squares. Use centimeter grid paper (PDF – be sure to print this document full scale) to determine the area of your handprint. What are the disadvantages of this method?
- Another method is to subdivide your handprint into sections for which you can easily calculate the area. Find the area of your handprint using this method. Does using the two methods result in the same area?

Up until now, you have been approximating the area of your handprint. In other words, your measurements were not exact.

**Problem A4**

What can you do to make your approximation more accurate? Explain why this approach will lead to a better approximation.

Another way to approximate the area of a handprint or any other irregular shape is to determine the number of squares that are completely covered and the number of squares that are partially covered. Average these two numbers to get an approximate area in the number of square units.

**Problem A5**

Think about the following statement: If you repeatedly use a smaller and smaller unit to calculate the area of an irregular shape, you will get a closer and closer approximation and eventually find the exact area. What do you think of this line of reasoning? Explain.

**Problem A6**

The palm of your hand is about one percent of your body’s surface area. Doctors sometimes use this piece of information to estimate the percent of the body that is affected in burn victims. Use your data to approximate the amount of skin on your body.

**Problem A1**

No, the surface area should be identical, since it is the fingers, not their location, that determine the area. The same shapes in different locations will have the same area.

**Problem A2**

- A good unit of measurement might be a square centimeter or a square millimeter on grid paper. An important aspect is that the unit should tile well.
- A circle does not tile well; circles leave holes in between them, and counting the area of the circles would not approximate the total area of the hand.

**Problem A3**

- Answers will vary. Counting squares is subject to several errors, including the possibility of miscounting the squares and the rounding error introduced by trying to count “half squares.” Additionally, it may not be the most precise method. It also takes a while to do.
- Answers will vary. Using two different methods will most likely result in different numerical values, since the area in both cases is an approximation and therefore subject to error.

**Problem A4**

Answers will vary. One useful method is to use paper with a finer grid, as smaller squares (units) will result in fewer rounding errors.

**Problem A5**

This is a reasonable method of approximation, although there are still other forms of measurement error that can have an effect on the calculation. We can increase the accuracy of our measurement by making the units smaller, for example, using mm^{2}instead of cm^{2} grid paper. But because we are physically measuring it, the area will always be an approximation, no matter how small the unit (and there’s always a smaller unit!). The measurement process always results in an approximate rather than an exact value.

**Problem A6**

Answers will vary. You can make this approximation by multiplying your palm’s area by 100.