How to Multiply by Geometric Constructions

Geometry can demonstrate the principles of arithmetic.

Geometric construction not only illustrates the principles of geometrical shapes but most arithmetic operations as well. The obvious way to illustrate multiplication is with the product of a rectangle's sides equaling its area. What's less obvious is constructing a segment as the product of two other segments, because multiplying lengths results in an area, not a distance. It only makes sense to do so if one of the lengths represents something other than distance. So, carrying out the construction requires defining an arbitrary unit to represent either a dimensionless quantity or a unit of area.

Things You'll Need

• Compass
• Straightedge

Instructions

1. Draw three line segments whose lengths correspond to two numbers to be multiplied and a unit of measure. Label the lengths corresponding to the numbers "s" an "t."

2. Draw an angle that measures about 60 degrees and mark the vertex "A." Make the length of the first leg of your angle at least s + 1 and the second at least t + s*t. For example, if you want to multiply 3 times 2, make sure the first leg is at least 4 times as long as the unit segment and the second is at least 8 times as long. If you need more room, make your unit segment shorter.

3. Expand the compass width to the length of your unit segment. Place the tip on point A and swing a small arc intersecting the first leg and mark it "B." Set the width to t and with the point on A, mark the other leg. Label the second mark "C" and connect it to point B with a straightedge.

4. Expand the compass width to the length s. Place the tip on point B and mark another point on the first leg on the opposite side from A. Label that point "D." The first leg should now have points A, B and D in that order.

5. Set the compass width to the whichever is shorter between BA and BC. With the compass point on B, make an arc intersecting the longer of the two segments. Mark that point "M." Imagine a new angle ADE equal to ABC with "E" being a point you will draw on the ray AC.

6. Place the tip of the compass with the same width on point D and swing an arc intersecting segment AD, making it large enough so it intersects where you imagine segment AE to be. Where the arc intersects AD, mark the point "N."

7. Set the compass width to the distance between M and either A or C, depending on where you drew M. Place the tip of the compass on N and mark a point of intersection with the last arc you drew. Draw a segment from D through the point of intersection and extend it to ray AC. Mark the end of the segment "E" where it meets the ray AC. Segment BC and DE should be parallel, and the length of segment CE should be equal to s*t.

Tips & Warnings

• This construction works because of similar triangles. Segment AB was defined as 1 unit in length, so 1/AC = BD/CE. Therefore CE = AC*BD.

References

• Iowa State Univeristy; Geometric Constructions; Lesley Lamphier; 2004
• Tom Davis; Classical Geometric Construction; Tom Davis; December 2002