Work out the Circumference of a Circle

The circumference of a circle is the distance around its edge. If a circle has a circumference of 2 miles (3.2 kilometers), you would have to walk 2 miles (3.2 km) around the circle before you came back to the place you started. When you're working on a geometric problem, though, you don't need to leave your seat. Read the problem carefully to find out whether it tells you the circle's radius (r), diameter (d), or area (A), then find the section that matches your problem. There are also instructions for finding the circumference of an actual circular object you want to measure.

Steps

Finding the Circumference from the Radius

  1. Draw a "radius" on the circle. Draw a line from the center of the circle to anywhere on the circle's edge. This line is the "radius" of the circle, often written as just r in math equations and formulas.
    • Note: if your math problem doesn't tell you the length of the radius, you might be looking at the wrong section. Check whether the sections for Diameter or Area make more sense for your problem.
  2. Draw a "diameter" across the circle. Extend the line you just drew so it reaches the circle edge on the other side. You've just drawn a second radius. The two radii stuck together have a length of "2 x the radius," written as 2r. The length of this line is the "diameter" of the circle, often written d.
  3. Understand π ("pi"). The π symbol, also written as pi. It isn't a magical number that just happens to work in this kind of math problem. Actually, the number π was originally "discovered" by measuring circles: if you measure the circumference of any circle (for instance with a tape measure), and then divide by the diameter, you'll always end up with the same number. This number is unusual because it can't be written out as a simple fraction or decimal. Instead, we can round to a "close enough" number like 3.14.
    • Even the π button on a calculator doesn't use the exact value of π, although it is close enough.
  4. Write down the definition of π as an algebra problem. As explained above, π just means "the number you get when you divide the circumference by the diameter." In the form of a math formula: π = C / d. Since we know the diameter equals 2 x the radius, we can also write this as π = C / 2r.
    • C is just a shorter way of writing "circumference."
  5. Change this problem so you are solving for C, circumference. We want to find out what the circumference is, which is C in this math problem. If you multiply both sides by 2r you get π x 2r = (C / 2r) x 2r, which is the same as 2πr = C
    • You might have written the left side as π2r, which is also correct. People like to move the numbers in front of the symbols just so the equation is easier to read, and this doesn't change the result of the equation.
    • In a math equation, you can always multiple the left side and the right side by the same amount and still end up with a correct equation.
  6. Plug in the numbers to solve for C. Now we know that 2πr = C. Look back at the original math problem to see what r (the radius) equals. Then replace π with 3.14, or use a calculator's π button to get a more accurate answer. Multiply 2πr together using these numbers. The answer you get is the circumference.
    • For example, if the radius is 2 units long, then 2πr = 2 x (3.14) x (2 units) = 12.56 units = the circumference.
    • In the same example, but using a calculator's π button for better accuracy, you'll get 2 x π x 2 units = 12.56637... units but unless instructed otherwise by your teacher, you can Round-Decimals the number to 12.57 units.

Finding the Circumference from the Diameter

  1. Understand what the "diameter" is. Put your pencil down on the circle's edge. Draw a line through the center of circle and hitting the edge on the other side. This line is the "diameter" of the circle, often written d in math problems.
    • The line goes through the exact center of the circle, not just anywhere within the circle.
    • Note: If the word problem doesn't tell you how long the diameter is, use a different method instead.
  2. Learn what d = 2r means. The "radius" of the circle, also written as r, is the distance halfway across the circle. Since the diameter extends all the way across the circle, the diameter is equal to two radii. A simple way to write this is d = 2r. This means you can always replace a d with a 2r in a math problem, or the other way around.
    • We'll be using d, not 2r, since your math problem tells you what d equals. However, it's important to understand this step, so you aren't confused if your teacher or math book uses 2r where you would expect a d.
  3. Understand π ("pi"). The π symbol, also written as pi, isn't a magical number that just happens to work in this kind of math problem. Actually, the number π was originally "discovered" by measuring circles: if you measure the circumference of any circle (for instance with a tape measure), and then divide by the diameter, you'll always end up with the same number. This number is unusual because it can't be written out as a simple fraction or decimal. Instead, we can round to a "close enough" number like 3.14.
    • Even the π button on a calculator doesn't use the exact value of π, although it is extremely close.
  4. Write down the definition of π as an algebra problem. As explained above, π just means "the number you get when you divide the circumference by the diameter." In the form of a math equation: π = circumference / diameter or π = C / d.
  5. Change this problem so you are solving for C, circumference. We want to find out what the circumference is, so we need to get C alone on one side. Do this by multiplying each side of the equation by d:
    • π x d = (C / d) x d
    • πd = C
  6. Plug in the numbers and solve for C. Look back at the original word problem to see what the diameter equals, and replace the d in this equation with that number. Replace π with an estimate such as 3.14, or use the π button on your calculator for a more accurate result. Multiple the values for π and d together, and you get C, the circumference.
    • For example, if the diameter was 6 units long, you'll get (3.14) x (6 units) = 18.84 units.
    • In the same example, but using a calculator's π button for more accuracy, you'll get π x 6 units = 18.84956... but unless instructed otherwise, you can Round-Decimals the number to 18.85 units.

Finding the Circumference from the Area

  1. Calculate-the-Area-of-a-Circle. Most of the time, people don't measure the area (A) of a circle directly. Instead, they measure the radius (r) of the circle, then calculate the area using the formula A = πr2. The reason why this formula makes sense is a little tricky, but you can find out more here if you are interested and willing to tackle some tougher algebra.
    • Note: If the math problem does not tell you the area of the circle, you may need to use a different method on this page.
  2. Learn a formula for calculating the circumference. The circumference (C) is the distance around the circle. Typically, you find it with the formula C=2πr, but because we don't yet know what the radius (r) is, we'll have to spend some time figuring out the value of r before we can solve it.
  3. Use the area formula to get r on one side. Since A = πr2, we can rearrange this formula to solve for r instead. If the steps below are difficult for you to follow, you might want to start on some easier algebra problems or Do-Algebra.
    • A = πr2
    • A / π = πr2 / π = r2
    • √(A/π) = √(r2) = r
    • r = √(A/π)
  4. Change the circumference formula using what you found. Any time you have an equation, such as r = √(A/π), you can replace one side of the equation with the other. Let's use this technique to alter the circumference formula above, C=2πr. For this problem, we don't know the value of r, but we do know the value of A. Let's change it like this to make the problem solvable:
    • C = 2πr
    • C = 2π(√(A/π))
  5. Plug in the numbers to find the circumference. Use the area given by the problem to solve for the circumference. For instance, if the area (A) of a circle is 15 square units, enter 2π(√(15/π)) into your calculator. Remember to include the parentheses.
    • The answer for this example is 13.72937... but unless instructed otherwise you can round to 13.73.

Finding the Circumference of a Real Circle

  1. Use this method to measure real circular objects. You can measure the circumference of circles you find in the real world, not just in word problems. Try it out on a bicycle wheel, a pizza, or a coin.
  2. Find a piece of string and a ruler. The string must be long enough to wrap around the circle once, and flexible enough that it can wrap tightly. You'll need something to measure the string with later, such as a ruler or tape measure. The string will be easier to measure if the ruler is longer than the piece of string.
  3. Wrap the string around the circle once. Start by placing one end of the string against the edge of the circle. Loop the string around the circle and pull it tight. If you are measuring a coin or other thin object, you might not be able to pull the string tight around it. Lay the circular object flat instead and arrange the string around it, as close to it as you can get.
    • Be careful not to wrap more than once. You should end up with a single loop of string, so there is no part of the circle with two lengths of string next to it.
  4. Mark or cut the string. Find the place on the string that completes the loop, touching the end of the string that you started with. Mark this place with a permanent marker, or use a pair of scissors to cut it at this points
  5. Unravel the string and measure it with a ruler. Take the loop of string and measure it on a ruler. If you used a marker, only measure from the end of the string to the colored mark. This is the part of the string that was wrapped around the circle, and since a circle's circumference is just the distance around the circle, you've found the answer! The length of this string is the same as the circumference of the circle.

Tips

  • You can write the plural of radius as either radii or radiuses.[1]

Related Articles

Sources and Citations

  1. ↑ http://www.merriam-webster.com/dictionary/radius
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