Blame Welsh mathematician William Jones for the trouble pi causes you. In 1706 Jones used the Greek word for circumference, περιϕέρεια, and shortened it to its first letter, π. As for π in circles, we have no one to blame but the marvel that is mathematics. Pi, or π, is the irrational number that links every circle's circumference to diameter and radius.

## Pi

Pi is an irrational number, meaning it cannot simplify to a ratio (a numerator and a denominator), and it cannot terminate (end) as a decimal. We can get close to π. Mathematicians would say they can "approximate" it, with either a fraction or a decimal:

• 22/7
• 14159

You can recall the right digits to five decimal places with this simple mnemonic: "Wow! I made a great discovery!" Check out the number of letters in each word: 3, 1, 4, 1, 5, 9.

In a circle, the circumference is related to the diameter by π. That relationship is a ratio:

• Circumference/Diameter

In every circle, of any size, that ratio is π. So we can use an irrational number -- a number that cannot be a ratio -- to establish a ratio. Some people say that is weird, but most mathematicians think it just shows the power of number.

## Getting Circumference from Diameter

Because circumference is related to diameter, we can use that ratio to find a missing value. Suppose you know the diameter of a circle but need to find its circumference. Circumference, C, can be placed in a ratio with diameter, d, with the answer pi, π. We can rewrite that relationship:

• C/d = π
• C = π x d
• C = πd

Here is a circle, the center of a soccer field. It is 10 yards in diameter. What is its circumference?

[insert drawing/animation of soccer field and center circle 10 yards in diameter]

• C = πd
• C = π(10 yards)
• C = 31.4159 yards

Here is a basketball court. Its foul line circles and center circle are each six feet in diameter. What is any circle's circumference?

• C = πd
• C = π(6 feet)
• C = 18.84954 feet

## Getting Diameter from Circumference

Since we know circumference and diameter are related by π, we can use π to find diameter if we know circumference. We do a bit of mathematics to isolate diameter, d, on one side of the equation:

• C = πd
• C/π = d

Dividing the circumference by π, then, will give us the diameter.

A Major League Baseball pitcher's mound is built inside a circle that is 56.54862 feet in circumference. What is the diameter of the pitcher's mound circle?

[insert drawing of a pitcher's mound]

• C = πd
• C/π = d
• 54862 feet/3.14159 = d
• 18 feet = d

One other key relationship, radius to diameter, can also give you circumference. Radius is one half of a circle's diameter. This means we can substitute radius, r, in our formulas, using the substitution property of equality. If two radii are equivalent to one diameter, we can substitute diameter with 2r.

We can also move values around thanks to the commutative property of multiplication. These four equations are equivalent:

1. C = πd
2. C = π2r
3. C = 2πr
4. C/π = 2r

The striking circle of a field hockey field is a quarter-circle with a 16-yard radius, a four-yard straight line, and then another quarter circle. The field has two of these striking circles. After using chalk to mark all the straight lines, you have only enough chalk left to make a line 111 yards long.

[insert field hockey diagram. Use https://www.athleticbusiness.com/images/pdf/field-hockey-specs.pdf for help]

Calculate the circumference of the circle and add the two, four-yard straight sections.

Think before you peek!

Consider that four quarter-circles is equivalent to one entire circle of 16 yard radius. Then add eight yards for the two straight sections.

• C = πd
• C = π2r
• C = 2πr
• C = 2 x 3.14159 x 16 yards
• C = 100.53088 yards

Add the two straight sections: 100.53088 + 8 yards = 108.53088 yards

You just barely have enough chalk!

### Next Lesson: Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.
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