![\"The](\"\/b\/imgs\/numbers_game\/tng_title.gif\")
While baseball may be a game of inches, golf, it’s been said, is a game of angles. It’s better to approach Augusta National’s 11th green from the far left edge of the fairway than, say, what used to be the far right edge of the fairway. It’s better to leave yourself an uphill putt than a downhill putt. If you play a fade, you’re better off teeing up on the right-hand side of the teebox.<\/p>\n
Yes, in addition to being a mental exercise, a stroll in the park, and one of the few solitary sports in the world, golf is a game of angles. This week in The Numbers Game<\/em> we take a look at some of those angles: how far offline can you start a three-foot putt and still expect to make it? How about a ten-foot putt? How hard is it to hit the green or fairway?<\/p>\n Let’s find out. If you know two things – an angle (θ) and a length or two lengths – you can calculate the other angles. After all, one of the angles is a given: 90°. The “o,” “h,” and “a” in “Sohcahtoa” stand for “opposite,” “hypotenuse,” and “adjacent.” The other letters correspond to sin, cosine, and tangent:<\/p>\n SOH<\/em> – s<\/strong>in(θ) = o<\/strong>pposite\/h<\/strong>ypotenuse This is all we’ll need to do today’s calculations. Let’s get started.<\/p>\n From the Tee<\/strong> How accurate does one have to be from the tee to find the fairway with a 250-yard drive? Let’s see:<\/p>\n In this example (and all others to follow), we cut the width of the target in half. A miss five yards to the right is the same as five yards to the left in terms of angle, so we’ll be expressing the range of angle as a +\/- number. The actual angular range is thus double the maximum angle itself.<\/p>\n Every example in this article will use a triangle similar to the one above. Since we have the opposite and adjacent sides for the given angle, the TOA<\/em> portion of Sohcahtoa becomes relevant: tan(θ) = opp\/adj. Plugging in our actual numbers, we get:<\/p>\n Solving for theta (θ) yields 4.004°. So, to hit a 35-yard wide fairway, a golfer has a “safety zone” of +\/- 4°. If he pulls or pushes his drive by more than these 4°, he’ll find himself in the rough.<\/p>\n How does the angle change on a wider – or narrower – fairway? This chart shows the results:<\/p>\n You’ll notice (for example, with the 20\/40\/80 trio) that as the fairway width doubles, so too does the angle. This makes sense, given that we’re using a simple fraction (20\/250, 40\/250, 80\/250) to calculate.<\/p>\n Tiger’s Driving<\/strong> Again, since we’re just using a simple ratio, at 300 yards Tiger must be 33% more accurate with his drive than someone who drives the ball 200 yards. Mixing the results, Tiger must also be about 34% more accurate than someone playing to a 35-yard wide fairway with a 250-yard drive.<\/p>\n Finally, if Tiger tries to hit one 350 to a fairway 28 yards wide, his angle comes out to 2.29°, or about 43% more accurate than your 250-yard drive on a 35-yard wide fairway.<\/p>\n Approach Shots<\/strong> In other words, from 50 yards out, your shank is still not going to find the green – you have only +\/- 14°. From 225, perhaps trying to reach a par-five green in two with a 3-wood or hybrid, that angle decreases to only +\/- 3.18°. If you found a 35-yard fairway with your 275 yard drive (+\/- 3.65°), you’ll have to be even more accurate trying to reach the green.<\/p>\n Putting<\/strong> For the purposes of examining putting, we’re clearly going to have to ignore lip-outs. The math here essentially assumes that every putt is struck the exact firmness to stop at the exact center of the hole. This has the effect of making the hole play as wide as possible, as a ball that is inside the hole by a millionth of an inch will still be made.<\/p>\n For the purpose of this chart, the Putt length is listed in feet and the hole size is listed in inches. Conversion is done prior to the calculation of θ.<\/p>\n While you may find the generosity of missing your intended line on a one-foot putt by +\/- 10°, the odds of making even a five-footer seem stacked against you. You have only a margin of +\/- 2.03° – again, with absolutely perfect<\/em> speed – to make the putt.<\/p>\n Dave Pelz is famous for saying thatt putts should be struck firmly enough travel 17 inches by the hole if they miss. I’m going to guess that this effectively makes the hole about 3½ inches wide – anything that catches the outer 3\/8″ of the hole is not going to go in at that speed.<\/span> A putt struck this firmly will only fall into the hole if it hits the middle 2.3 inches! With this little correction in mind, the putting chart becomes really<\/em> daunting.<\/p>\n Want to make a 100-foot putt? You’d better get the line right – down to +\/- 0.055° – with near-perfect speed. With these numbers, you’ve gotta wonder how we ever make even a three-foot putt let alone the occasional putt from ten feet.<\/p>\n Golf is a Hard Game<\/strong> Imagine telling someone who has never heard of golf that you’ve invented a sport that involves hitting a ball with oddly shaped sticks into a teeny little hole placed 400 yards away. How many times, you could ask the person, do they think you’d have to hit the ball to achieve the goal. 50? 100?<\/p>\n The fact that we can get the ball into the hole in even eight strokes – let alone four, three, or even two on rare occasion – is a true testament to how good we really all are at golf. So, the next time you’re furious about your 85, think back to this article and the little story above. It may not improve your game, but it does offer a little consolation.<\/p>\n","protected":false},"excerpt":{"rendered":" It’s a wonder anyone makes putts longer than two feet given this math!<\/p>\n","protected":false},"author":25,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[24],"tags":[],"class_list":["post-1828","post","type-post","status-publish","format-standard","hentry","category-the_numbers_game"],"yoast_head":"\n
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\nSohcahtoa<\/strong>
\nVirtually all the math in this edition of The Numbers Game<\/em> is going to involve right triangles, and Sohcahtoa can help. Sohcahtoa is not an Indian name, but a mnemonic device to help you remember how the math works in calculating angles and lengths of right triangles. What’s a right triangle? Well, it’s a triangle in which one of the angles is “right” – perpendicular or 90°.<\/p>\n![\"Right](\"\/b\/imgs\/numbers_game\/right_triangle.jpg\")
In this right triangle, I’ve marked two angles and color-coded their adjacent and opposite sides. The hypotenuse is always the longest side.<\/p>\n
\nCAH<\/em> – c<\/strong>os(θ) = a<\/strong>djacent\/h<\/strong>ypotenuse
\nTOA<\/em> – t<\/strong>an(θ) = o<\/strong>pposite\/a<\/strong>djacent<\/p>\n
\nThe average fairway width at most golf courses is approximately 35 yards. The U.S. Open sets its fairways to 22-28 yards in width, and I have a hole at my course – a par five no less – that features a fairway measuring only 17 yards wide.<\/p>\n![\"Right](\"\/b\/imgs\/numbers_game\/right_triangle_250yd_drive.jpg\")
I will not be drawing the triangles to scale, so please ignore the fact that the 17.5-yard side is about a third as large as the 250-yard side.<\/p>\ntan(θ) = 17.5\/250 = 0.07<\/pre>\n
Drive Width tan(θ) θ (+\/-)\r\n----- ----- ------ -------\r\n250 17 0.03 1.95\r\n250 20 0.04 2.29\r\n250 25 0.05 2.86\r\n250 30 0.06 3.43\r\n250 35 0.07 4.00\r\n250 40 0.08 4.57\r\n250 45 0.09 5.14\r\n250 50 0.10 5.71\r\n250 65 0.13 7.41\r\n250 80 0.16 9.09\r\n250 100 0.20 11.31<\/pre>\n
\nMuch has been made of Tiger’s inability to find the fairway, but the simple truth is that it’s tougher to find the fairway 300 yards from the tee than it is 200 yards from the tee. How much harder? Let’s re-run the calculations, this time keeping the fairway width at 28 yards (the average fairway width on the PGA Tour, and a smaller fairway than you’re likely used to hitting to) and changing the drive’s yardage. Doing so, we get:<\/p>\nDrive Width tan(θ) θ (+\/-)\r\n----- ----- ------ -------\r\n200 28 0.0700 4.00\r\n210 28 0.0667 3.81\r\n220 28 0.0636 3.64\r\n230 28 0.0609 3.48\r\n240 28 0.0583 3.34\r\n250 28 0.0560 3.21\r\n260 28 0.0538 3.08\r\n270 28 0.0519 2.97\r\n280 28 0.0500 2.86\r\n290 28 0.0483 2.76\r\n300 28 0.0467 2.67<\/pre>\n
\nImagine a rectangular green 25 yards wide, or 75 feet edge to edge. The depth is irrelevant – we’re going to be shooting at the flag. How accurate must your approach shot be to find a piece of the green?<\/p>\nAppr. Width tan(θ) θ (+\/-)\r\n----- ----- ------ -------\r\n 50 25 0.2500 14.04\r\n 75 25 0.1667 9.46\r\n100 25 0.1250 7.13\r\n115 25 0.1087 6.20\r\n130 25 0.0962 5.49\r\n150 25 0.0833 4.76\r\n175 25 0.0714 4.09\r\n200 25 0.0625 3.58\r\n225 25 0.0556 3.18<\/pre>\n
\nA 25-yard green is going to look awfully big compared to a 4¼-inch hole!<\/p>\nPutt Hole tan(θ) θ (+\/-)\r\n---- ---- ------ -------\r\n 1 4.25 0.1771 10.04\r\n 3 4.25 0.0590 3.38\r\n 5 4.25 0.0354 2.03\r\n 10 4.25 0.0177 1.01\r\n 15 4.25 0.0118 0.68\r\n 20 4.25 0.0089 0.51\r\n 25 4.25 0.0071 0.41\r\n 30 4.25 0.0059 0.34\r\n 50 4.25 0.0035 0.20\r\n100 4.25 0.0018 0.10<\/pre>\n
Putt Hole tan(θ) θ (+\/-)\r\n---- ---- ------- --------\r\n 1 2.3 0.09583 5.474\r\n 3 2.3 0.03194 1.830\r\n 5 2.3 0.01916 1.098\r\n 10 2.3 0.00958 0.549\r\n 15 2.3 0.00639 0.366\r\n 20 2.3 0.00479 0.275\r\n 25 2.3 0.00383 0.220\r\n 30 2.3 0.00319 0.183\r\n 50 2.3 0.00192 0.110\r\n100 2.3 0.00096 0.055<\/pre>\n
\nI used to get pretty down on myself when playing poorly. A few years back I thought of something.<\/p>\n