Trigonometry Averts Loss of Airplane
By: Michael Lodato
On a dark and stormy night in late December 1955 I was on duty at a USAF radar base in north-eastern Italy near the town of Campoformido.
Just north of us were the Dolomite Mountains, also known as the Italian Alps, but it was fairly level around our base at an elevation of 300 ft. The cloud cover (ceiling) was at best 1500 feet above us. We got a call from a USAF pilot in a C-47 who told us he was in a storm, low on fuel and needed help. We were not really in position to give it to him. We didn’t even have a paved landing strip. We had a C-47 on our base so that fliers could log in time and earn their flight pay, but they took off and landed on a grass field – only in the day time because there were no landing lights.
We did have a search radar for detection on aircraft (see photo). It rotated steadily, sweeping the airspace with a narrow beam. When the beam struck a flying object, a blip appears on a circular radar scope. We also had a second radar. It didn’t rotate but did bob up and down in the direction it was pointed and gave the altitude of the flying object at which it was pointed.
My goals were to
- Get the plane to travel in the right direction – toward us.
- Get enough lighting so the pilot could complete a safe landing,
- Tell the pilot at what altitude he should be at different distances from us as he approached the landing.
Since the grass landing strip was right next to the radar, direction was not a problem – just aim the plane (blip) at the center of the radar scope, and watch the blip move in the right direction.
For lighting I asked a sergeant to get all the trucks and cars he could find, line them up on both sides of the grass runway and turn their lights on. This worked very well.
Goal number 3 was harder. But I came up with a crazy idea. A year earlier I had earned a degree in mathematics, and I remembered some things from trigonometry that might help. I remembered that ratios of the sides of a 30-60-90 degree (right) triangle were as follows: if the short leg (opposite the 30 deg. angle) is of length L, the diagonal (hypotenuse) is of length 2 times L and the long leg (opposite the 60 deg. angle) is L times the square root of 3 – in symbols √.3
So for different ground distances, D, from the landing point, I had to calculate the altitude, A, (desired altitude) in feet at which the plane should be on its descent and feed that info to the pilot so he could adjust his altitude accordingly. In trigonometry the equation is:
Distance from base (D) = Desired altitude (A) times √3
For example, at 10 miles from the base I had to solve for A in the equation:
10 (miles) = A (miles) × √3 or A = 10 miles ÷ √3
Or A = 10 miles ÷ 1.732 = 5.7735 miles
To get the altitude in feet, you multiply by 5,280 – the number of feet in a mile. Hence the altitude should be 30, 485 feet at 10 mile from the base.
I knew these numbers wouldn’t work because a C-47 couldn’t fly above about 25,000 feet and a 30 degree landing angle might be too steep. So I decided to use 50% A as the desired altitudes. So at 10 miles out the plane should be at about 15,000 feet.
The plane’s altitude at 5 miles out should be = ½ x 5 ÷ 1.732 = 1.44 miles or about 7,600 ft.
The plane’s altitude at 2 miles out should be = ½ x 2 ÷ 1.732 = .58 miles or about 3,000 ft.
This is simple arithmetic today – my hand calculator gives me the square root of a number with one click. But in 1955 we had no calculators. To get the square root of 3, we started with a guess, multiplied the number by itself, looked at the result adjusted the guess up or down and did it again. After doing this multiple times we set the square root of 3 at 1.732.
The result of all of this was what I thought was a miracle – the plane burst through the cloud cover in line with the lights provided by the cars and trucks. Everybody was delighted especially the pilot and his passengers that included a USO troupe of musicians and dancers. Lucky us. We all went to a large villa that I with 3 other officers rented and had a grand party.
About a year later I was dining in the Officers’ Club at a base outside of Casablanca and the pilot of that flight saw me from across the room and brought a group of people over and related the story to them.
In World War II our site was an Italian air base. The place we used for the Officers’ Club was the basement of a building that was bombed in December 1945. It turns out that one of my golfing buddies may have been the one that dropped the bombs. On the record of his missions was a note that, on Christmas Day 1945, while returning from escorting bombers to and from the Ploesti oil fields in Romania in his P-38 fighter, he dropped some bombs on the Campoformido air base.
Wow! I love math and it is great to see how you used it “back in the day”. Life or death situation.
Tim Foster says
I remember sharing this with my Trigonometry professor, he never forgot it and I’ve heard that he still tells the story of this each semester of how Trigonometry can be applied in a life or death situation. It makes me smile. You have had a life full of pretty incredible stories. This will always be one of my favorites.
Michael Lodato says
Stay tuned, Tim. There are a lot more stories.