# Hockey Geometry

From the passes NHL players make to their teammates, to the shots they take to score, players in every position are constantly using geometry when playing the game. The lines, angles and curves on the ice are also examined. "Science of NHL Hockey" is a 10-part video series produced in partnership with the National Science Foundation and the National Hockey League.

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**LESTER HOLT, reporting:** See it? There, at the face-off, when the puck is dropped. There, dividing the zones of play. There, in the pass off the boards. Where, in an NHL hockey game or on an NHL rink do you see geometry?

**Dr. EDWARD BURGER (Williams College):** We see things like circles, squares, rectangles. There are points, there are lines, there are angles between two, two rays. The hockey rink is teeming with geometry.

**HOLT:** Especially plane geometry

**BURGER: **Plane geometry is a study of things that are flat , like triangles, squares, circles and so forth

**HOLT:** Like the face-off circles, one at center ice, and two in each end zone. To start or resume play, an official drops the puck into the center of the circle, between two opposing players. The geometry of a circle gives each player an equal chance of getting the stick on the puck: a circle is a closed curve, with all points on the curve the same distance from the center.

**BURGER:** And the distance away from the center point out to that boundary is called the radius.

**HOLT:** The radius of an NHL face-off circle is 15 feet.

**BURGER:** If you double the radius that is going to be the diameter, which goes from one end of the circle all the way to the other and passes through the center. That’s the diameter.

**HOLT:** Now look at the two blue lines that run across the ice, dividing it into three zones. If an attacking player crosses the center, or ‘neutral’ zone and the blue line marking the start of the other team’s zone before the puck crosses it, that player is offside, one of the most common violations in the game. The neutral zone is another common geometric shape – a quadrilateral or four-sided shape. Since it has two pair of parallel sides, it’s also a parallelogram. And since the sides are at right angles (more on that later), it’s a rectangle. Other geometric shapes are painted on the NHL rink, a semicircle in front of the scorekeeper, truncated’ semicircles (half-circles with the ends cut off) in front of each net, called ‘goal creases’ – when the goaltender is in this area, attacking players must not obstruct his movement or vision. Goalies can also play the puck in this area, behind the net, called the ‘goaltender trapezoid.’ Trapezoids are quadrilaterals, too, but with only one pair of parallel sides. As for the puck itself, it’s an example of solid or 3-dimensional geometry: take a circle…add the dimension of height, and you get a geometric shape called a cylinder. Now let’s take a look at all of this from another angle: angles a fundamental part of hockey and geometry. What is an angle? Think of two straight lines or ‘rays’ that start at the same point. Now imagine a circle laid over it, with the center of the circle at the point. The angle measures the arc – the turn – between the two lines, in degrees.

**BURGER: **One complete cycle around is 360 degrees. If you just go half around, that 180 degrees. A quarter of a circle would actually be a quarter of 360, which of course is 90 degrees, and that’s called a right angle.

**HOLT:** An angle the shape of the corners of a square or a rectangle, and the easiest angles to see on a hockey rink. Angles that are larger than 90 degrees (but smaller than 180 degrees) are called obtuse angles. Good example? What’s called the lie of the hockey stick – the angle between the blade and the shaft ,which can vary, but is often around 135 degrees. Angles smaller than 90 degrees are called acute angles, which are best visualized (for those with quick eyes), during play, watching defenders try to keep attackers from scoring by limiting their ‘angle of access’ or ‘angle of attack’ to the net. An attacker directly in front of the net has the widest openings on either side of the goalie to shoot the puck through.

**BURGER:** But suppose that this guy has some team players that pushes this person off. Now from the point of view of shooting, the angle of access is smaller, which means it’s going to be harder to make that shot.

**MATT MOULSON (Left Wing, New York Islanders):** The goalie is obviously thinking about angles all the time, judging where shooters are shooting from and the angles they have to score..

**HOLT:** All players are visualizing, and planning and anticipating, angles of puck movement, so they can, with precision, pass the puck…or ‘ricochet’ it – ‘bank it’ – off the sides of the rink, called the angle of incidence and the angle of reflection, which are equal, meaning the puck tends to bounce off the boards at the same angle it hits the boards.

**ERIK JOHNSON (Defenseman, Colorado Avalanche):** If you are rushing the puck up the ice and you can’t make a direct pass to a forward, you got to put it in to the boards at an angle and have them skate to it. So that’s another thing where we use angles and put the pucks in certain areas. I wasn’t very good at geometry in school, but on the ice you actually use it a lot more than you think.

**HOLT:** And use it blink-fast.

**BURGER:** Those players organically, actually, are thinking mathematically even if they’re not doing it consciously. And you see it on the ice. They’re great players because they are great mathematicians.