A Carpenter Draws a Circle

 
Objective: To demonstrate how to draw a circle using a carpenter's square, also known as a steel, framing, or rafter square.

Level: This demo is appropriate post-algebra classes in geometry, trigonometry, introductory modeling, or possibly a general mathematics class for industrial arts.

Prerequisites:
<>
For a geometry or an industrial arts class students should be familiar with circles, the Pythagorean Theorem, and completing the square.
<> For a trigonometry class students should be familiar with sine and cosine in right triangles, the Pythagorean identity, and properties of circles.
<> For an introductory modeling class students should be familiar with properties of circles, trigonometry, the Pythagorean identity, and parametric representation of a plane curve.

Platform: No electronic technology required. However, we include animations, an Excel routine, and a Geometer's Sketch Pad routine for visualization purposes.

Materials Required: A carpenter's square; a piece of paper, poster board, or cardboard; wood or heavy cardboard for a backing; push pins for the ends of a diameter, and thumb tacks or push pins to hold the paper to the backing; pencil or pen.

Instructor's Notes:

Background: Common tools often conceal many levels of usefulness. One such tool is a standard metal carpenter's square. It is constructed of flat steel or aluminum and is L shaped. Figure 1 shows carpenter's squares of several sizes and styles.

(a) A small square; outside dimensions 8 in. by 12 in.

(b) A larger square etched with tables of rafter lengths;
outside dimensions 16 in. by 24 in.

(c) A small adjustable square with a level, often used for marking material for straight cuts; 9 in. long.

(d) An older style adjustable square; 12 in. long.

Figure 1.

Using a square to draw a circle: It used to be the case that carpenters didn't know much mathematics beyond the ability to read a tape measure and do simple computations. To draw small circles they may have used a compass but for larger circles they may have used their carpenter's square in the manner we describe next.

  • A fix large piece of paper or poster board to a piece of wood that acts as a backing. (You can use thumb tacks, push pins or tape.)
  • Insert two push pins into the drawing surface; press the pins in firmly, but not all the way in. (See the green pins in Figure 2.) The distance between these pins should be smaller than the length of the short leg of square. In Figure 2 the pins are about 7 inches apart.
  • Lay the square so that each leg is against the base of the push pins. (Here we are using the small square as shown in Figure 1(a).
     

    Figure 2.

     

  • With your pencil trace along the top edge of both legs of the square. Put a dot at the top corner of the square. Figure 3 displays the result of this step when the square is removed.

Figure 3.

  • Keeping the legs of the square against the push pins rotate it counter clockwise a bit. Now with your pencil trace along the top edge of both legs of the square. Put a dot at the top corner of the square. See Figures 4(a) and 4(b).

Figure 4(a)

Figure 4(b)

  • Repeat the previous step to generate a series of points that start at the right push pin and head toward the left push pin. See Figure 5.

Figure 5.

  • Connect the dots you put at the top corners as the square was rotated with a smooth arc. The result is a semicircle whose diameter is the distance between the push pins. (We need to prove this statement. See Mathematical Connections below.) The more triangles that you construct, the easier it will be to draw a smooth arc to join the dots.
     

  • To get the bottom half of the corresponding circle, once you near the left push pin flip the square so that its corner is pointing downward as shown in Figure 6  and repeat the construction steps.

Figure 6.

 

In the classroom:
<> For geometry or industrial arts classes to use in-class participation have a team of two students construct a semicircle or a full circle using the carpenter's square method. Have each team choose one of the triangles in their construction formed by connecting push pins as shown in Figure 5 and the tracings along the legs of the carpenter's square. Measure the lengths of the legs and the hypotenuse. Next use the Pythagorean theorem to show the triangle is a right triangle or very close to a right triangle.

<> For a trigonometry class to use in-class participation have a team of two students construct a semicircle or a full circle using the carpenter's square method. Have each team choose one of the triangles in their construction formed by connecting push pins as shown in Figure 5 and the tracings along the legs of the carpenter's square. Using a protractor have the students measure one of the angles formed by the hypotenuse and a leg and measure the length of the hypotenuse. Using sine and cosine formulas determine the lengths of the legs of the triangle. Next use the Pythagorean theorem to show the triangle is a right triangle or very close to a right triangle.

<> For a modeling class to use in-class participation have a team of two students construct a semicircle or a full circle using the carpenter's square method. Have each team choose one of the triangles in their construction formed by connecting push pins as shown in Figure 5 and the tracings along the legs of the carpenter's square. Label one of the acute angles of the triangle and measure the length of the hypotenuse. Using sine and cosine formulas determine expressions the lengths of the legs of the triangle. Next use the Pythagorean theorem to show the triangle is a right triangle or very close to a right triangle.

Animations and Interactive Excel routine: To see an animations of the construction procedure outlined above choose one of the following:

(For a free Quicktime movie player go to http://www.apple.com/quicktime/download/ )

To download or execute an interactive Excel routine for generating the semicircle click here. In this routine you can choose the diameter. Choosing the diameter between 1 and 12 generates a nice display.

To download or execute a Geometer's Sketchpad animation for generating a full circle click here. (Execution requires the Geometer's Sketchpad software.)

Mathematical Connections: Using the carpenter's square provides a technique for drawing a circle that doesn't rely on a compass. We must prove that the dots at the intersection of the legs of the carpenter's square are on the circumference of a circle.

Figure 7 will be helpful in order to show that the dots lie on a circle. At vertex C is a dot with coordinates (x,y) constructed using the carpenter's method. For convenience, we give the coordinates of the push pin at A as (0, 0 ) and those at the push pin at B as (d, 0), where d is the distance between the push pins.

Figure 7.

<> In a geometry or industrial arts class to prove that the dots recorded at the corner of the carpenter's square lie on a circle we can use the diagram in Figure 7. We proceed to find the equation of the circle containing the dots. Hint: use the Pythagorean theorem to determine the legs of triangle ACB and apply it again to the triangle ACB. Students may need an assist in developing the requisite lengths. For more details click here.

<> In a trigonometry class to prove that the dots recorded at the corner of the carpenter's square lie on a circle we can use the diagram in Figure 7. Label angle BAC as and determine the lengths of sides AC and BC in terms of trigonometric functions of and the length of the hypotenuse. Note that the dashed line from vertex C is a perpendicular to segment AB. This gives us two other right triangles. Use these triangles and trigonometry to determine formulas for x and y. Finally show that (x, y) lies on a circle by starting with x2 + y2, substituting in the trigonometric expressions for x and y, and simplifying to obtain the equation of a circle. Students may need an assist in developing the trigonometric expressions for x and y. For more details click here. An alternative proof using similar triangles is also included.

<> A project for a modeling class can focus on developing a parametric expression

for the set of dots that are connected to form the semicircle. The development in this case uses the same approach as given for the trigonometry class. After it is shown that the points (x,y) lie on a circle the derived parametric representation can be used to plot the set of points. This is how the animations mentioned above were developed. Click here for more details.

Auxiliary resources:

1. Jim Loy at http://www.jimloy.com/cindy/carp.htm has an interactive Java demonstration for using a carpenter's square to generate a circle. The program was created with Cinderella (a geometry program). The routine doesn't leave a track of the dots as is done in our animation and Excel routine.

2. Carpenter square have been used for a very long time. Click this URL for an interesting historical square http://www.canterburytrust.co.uk/schools/gallery/gall07g.htm

3. For a middle school activity involving a carpenter's square and square roots go to http://pumas.jpl.nasa.gov/examples/layout.asp?Document_Id=11_07_00_1

4. For some general uses of a carpenter's square in building see http://www.bobvila.com/ArticleLibrary/Task/Building/CarpenterSquare.html

5. For information on how a carpenter is taught to use a square in construction, see Carpenter's and Builders Library No. 1, Fourth Edition, by John E. Ball, published by Theorode Audel & Co., 1978, pp. 229-268.

Credits:  This demo was submitted by Sean Comfort, an undergraduate mathematics major at Temple University, who, in a continuing life, is also a skilled carpenter. The animations, figures, and accompanying text were constructed by David R. Hill, Temple University and Lila F. Roberts, Georgia College & State University. This demo is included in Demos with Positive Impact with their permission.


DRH 1/31/2005                              Last updated 9/15/2010 DRH

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