Colorado Public Radio (NPR) reports that a group of students taking an agriculture class at Soroco High School in Oak Creek, Colo., wanted to build a giant steel bale feeder. You know, the type of thing used when feeding horses, sheep, and other animals on a farm with bales of hay. You know the kind of thing I mean, right? A bale feeder.
Unlike typical readers of these pages, the students at Soroco probably do know what a bale feeder looks like: many of them live on ranches owned by their families. They also don’t seem to have much of a problem using an arc welder to build a bale feeder.
Where they have an issue is with the math. See, the steel used in the bale feeder costs $2.75 a foot, and if their calculations are off and they take a band saw to the steel and cut it in the wrong place or at the wrong angle, it’s like throwing away money. So, the calculations have to be right and they have to use the band saw with precision.
For reasons only ranchers seem to know, bale feeders are shaped like a regular octagon. They have to be a little bigger than the average bale of hay, which has a diameter of about 6 feet, I hear from the math teacher on the scene, Maggie Bruski, who posted that information in a comment on the NPR story. Students decided to make the diameter of the bale feeder 7 feet to accommodate the bales of hay, making the radius of an inscribed circle 3½ feet.
Given those requirements, students also knew a few other measurements:
- Given that a regular octagon has 8 sides of the same length, it’s made up of 8 isosceles triangles with one angle measuring 45° and the other two measuring 67½°.
- If we bisect one of those triangles with a line perpendicular to the octagon’s side, the resulting right triangle, shown in white, has an angle of 22½° at the center of the octagon.
Determining the length of steel to cut
To determine the length of a side of the hexagon, shown in red on the diagram above, we can use the aforementioned bisected isosceles triangles.
But we only know one of the legs, because we forced that to have a length of 3½ feet, or 42 inches. The other leg and the hypotenuse of our right triangle have lengths that are unknown to us at this point.
Here’s where trigonometry comes in to save the day for these students and their horses. Knowing that the tangent of an angle in a right triangle is equal to the length of the side opposite that angle divided by the length of the side adjacent to that angle, neither of which is the hypotenuse, we know that the length of the other leg, in inches, is given by
That comes out to 17.397 inches, approximately. Then, we also know that the length of the octagon’s side, which represents the length of steel the students need to cut, is twice that, or 34.794 inches, or about 2 feet 10¾ inches, they figured.
Determining the miter angle
Next, we have to remember that these eight equal-length pieces of steel, after they’re cut, will be welded together end to end and the last one welded back to the first one.
Zoom in now on the part of the diagram where the angle is labeled 67½°. Each internal angle of a regular octagon is 135°, according to a colorful lesson on Cool Math.com. To do this most efficiently, we should make each piece of steel contribute half of that angle, which comes to 67½ degrees each.
We therefore set the miter angle at 67½° and cut. When the two pieces of steel are welded together after the miter cut, they will form a 135° angle at every vertex of the octagon.
And finally, the horses can eat.
