Geometry at Work

Each year, I am tasked with building props for my wife’s dance studio, NH Academie of Dance. This year, I had to build two giant hearts that were at least 10′ tall each. The original plan was that these hearts would consist of a wooden border about 6″ wide with white fabric stretched across the middle.

First, I decided that I was going to use MDF (Medium Density Fibreboard) for the heart border. MDF is often used for making furniture, school projects, etc., and it seemed like it would be a good choice for my needs. Strong, yet flexible.

The next step was to draw the heart to scale. I needed to be able to recreate the hearts on a large scale, so just drawing it freehand wasn’t really an option. It was time to dust off my high school geometry knowledge. Using a protractor and 45 degree triangle, I drew a half circle with a 2.5″ radius on a 45 degree angle (my original plans were hand drawn, but I’ve included some images below that I recreated on the computer for this post… the originals seemed more impressive to me… there’s just something about getting highly precise drawings by hand that looks more impressive). This would be the top left side of the heart.

Next, I created a mirror image of this by creating another half circle on a 135 degree angle. This angle was the tangent created at the rightmost edge of the first half circle. This gave me the top right side of the heart.

Next, I extended the tangents on the outer edge of each circle until they intersected to form the bottom of the heart.

I needed to know the lengths of the bottom pieces of the heart, so I took a look at what I knew already.

1. The angles where the tangent lines crossed were all 90 degree angles so when dividing the heart in half down the middle, I get two right triangles, each with 45 degree angles for both of its complementary angles.
2. If the complementary angles of a right triangle have the same measure, then the legs of the triangle opposite those angles have the same length. I knew that the radius of my arc was 2.5″, which meant that the diameter was 5″. Since the two 45 degree angles are equal, the lengths of both legs must therefore be 5″. I now had a scale version of my heart shape.

Looking back at it, I now see a much easier approach that I could have taken. I could have started by drawing a square, then draw the arcs using half the length of one side of the square as the radius, from the midpoint of one of the sides. Then I could have rotated the whole thing 45 degrees and it’s the same heart.

At this point, I had the basic outer edge of my heart shape. There was to be some overlap on the bottom (giving it an almost ribbon-like appearance), and at the top-middle of the heart where the arcs met. For the top overlap, I simply extended each arc along it’s tangent for about 9″ (since the width of the border was 6″, that gave about 3″ of overlap). For the bottom, I started out by creating an arc whose center was on one of the tangent lines (extending the tangents farther down), but in the end I decided that those could be a bit more freehand.

I was quite happy with the end result, and glad I still remembered some of those Geometry lessons from high school.