Similar triangles

That’s the question I pose to aspiring private and commercial pilots who come to the Sewanee-Franklin County Airport (UOS) for their practical exam.

After obtaining the current atmospheric conditions and consulting the takeoff performance chart in the airplane’s operating handbook, an applicant might report that she needs 1,000 feet for the ground roll and 1,600 feet to clear a 50-foot obstacle. “With 3,700 feet of runway available, that will work out fine,” she says. My next question, “Great! Now, given that the trees at the end of the runway are closer to 100 feet tall, how much runway will we need for that?” is typically met with a long period of silence. And with good reason—there’s no one way to answer the question since the POH remains silent on clearing objects of varying heights. Still, pilots need to be able to come up with some reasonable answer to ensure takeoff safety. This question initially stumps even flight instructor candidates.

The illustration (right) demonstrates one way to tackle the problem. Think of the aircraft rotating after traveling 1,000 feet down the runway and climbing at a constant angle afterward using VX, best climb angle airspeed. After another 600 feet, the POH predicts the aircraft will be 50 feet in the air. If that climb angle stays the same, then after an additional 600 feet, or at the 2,200 feet mark along the runway, the aircraft should be 100 feet in the air. These nested right triangles are similar; they share the same angles. So, doubling the horizontal distance means the vertical distance follows suit. “Oh, I remember similar triangles from high school, but I never thought I’d use them again!” along with a chuckle, is a response I sometimes hear.

One caution about this technique is that it involves extrapolating from data in the pilot’s operating handbook and, although that is not always justified, we’ll see that it is in this case. The numbers in the above example come from the POH for my Beechcraft Bonanza E33C. Another reasonable way to solve the problem is to convert the climb rate provided by the POH into a climb angle for the second part of the climb from 50 feet to 100 feet. But the climb chart assumes we use best climb rate airspeed VY instead of VX. Still, the chart predicts a more favorable climb gradient than the extrapolation method, presumably because the takeoff performance chart accounts for an initially drag-intensive flight profile since the gear will be retracting during that segment. Given that we are trying to guarantee that the runway is long enough, I prefer the more pessimistic, and simpler, prediction extrapolating from the takeoff performance chart.

Once we have an estimate of the horizontal distance to clear the trees, we may want to add an additional distance if, by previously connecting takeoff estimates with actual performance, the POH predictions prove optimistic. For example, if the Bonanza generally uses 10 percent greater distance than book values, the new estimate to clear a 100-foot obstacle is 1.1(2,200) = 2,420 feet. Finally, it’s important to add another buffer to account for changes in winds, suboptimal technique, and so on. I like to add at least 50 percent to the final value to ensure that we have at least 1.5(2,420) = 3,630 feet of runway available. So, Sewanee’s 3,700-foot runway is barely long enough, in the given scenario, for a departure with a comfortable safety margin.

Surprisingly, practical exams for new certificates aren’t the only conversations where geometry keeps coming up. Applicants who add the instrument rating to a current certificate must all complete a circling approach, and knowing your way around similar triangles can prove useful.

An instrument approach involves a circling maneuver if the aircraft needs to change direction significantly from the final approach segment before landing on the runway. Such a circling approach turns out to be one of the sketchiest parts of an instrument practical exam—and for good reason. When flown low to the ground, in poor weather conditions by a tired pilot at the end of a long flight, it would be easy to make a mistake that proves fatal. Pilots should always consider other options to conduct a straight-in approach before choosing to circle.

It makes sense that each instrument candidate must demonstrate circling approach proficiency on the practical exam. Even though the exam is often conducted in beautiful VFR weather, the circling approach is one of the more common reasons I end up issuing a notice of disapproval. The Instrument Rating Airmen Certification Standards (FAA-ACS-8C) states that, “If landing, initiate a stabilized descent. Touch down on the first one-third of the selected runway without excessive maneuvering, without exceeding the normal operating limits of the airplane, and without exceeding 30 degrees of bank.” It’s easy to argue that a significant overshoot of the runway centerline, and the attendant maneuvering back to it, fails to provide a stabilized descent. During the preflight briefing, I remind that significantly overshooting centerline during the circling approach will result in a notice of disapproval. The candidate often says, “Oh, I will maintain a normal VFR pattern.” That’s great news and the right idea since a standard VFR downwind leg keeps the aircraft well within the maximum circling distance while minimizing the risk of an overshoot. Still, they routinely happened anyway, so I found myself scratching my head. I then realized that even though candidates thought they were flying a normal traffic pattern, they were using the wrong reference to maintain it (right). For example, on the downwind leg of a typical 1,000-foot pattern altitude, keeping the runway in the middle of the strut on my Cessna 152, Wilbur, means that I’m about one-half mile away from the centerline. In my Beechcraft Bonanza, Niky, it’s the leading-edge section that resides near the fuel cap that provides the appropriate reference. But if the circling altitude is 500 feet agl, half of normal, and I maintain the same reference, it means I’m half the normal distance to the runway.

Safely turning onto base and final within one quarter of a mile is nearly impossible if the maximum bank angle, as stated in the ACS, stays less than 30 degrees. For circling maneuvers, we need to find a new reference to maintain the normal distance from the runway on the downwind. In Wilbur, it’s about three-quarters of the way up the strut and in Niky, the reference is just outside the large vortex generator that protrudes forward close to the wing tip.

I enjoy mixing mathematics and aviation to understand what makes airplanes fly. But when I conduct a practical exam, I plan to keep discussions centered on aviation. Still, the topic of similar triangles kept coming up and sharing the topic outside of my practical exam debriefs I hope proves helpful for others.

Catherine Cavagnaro teaches aerobatics at UOS and is the Gaston Swindell Bruton Professor of Mathematics at Sewanee: The University of the South.