The circumference of a circle is the set of points equidistant from its center that delimit its area. If a circle has a circumference of 3 km, it means that you will have to walk that distance, along the entire perimeter of the circle, before you can return to the starting point. When you are struggling with geometry problems, to find the solution you will not need to leave the house to experiment physically. First read the problem text very carefully to identify the fundamental data of a circle, such as the radius (r), the diameter (d) or the area (A), then refer to the appropriate article section to find the solution to your specific problem. This guide also provides instructions for physically measuring the circumference of a circular object.
Steps
Method 1 of 4: Calculate the Circumference Using the Radius
Step 1. Draw the "radius" of a circle
Draw a line that starting from the center reaches any point on the circumference of the circle. The segment you drew represents the "radius" of your circle. Normally the radius is indicated with the letter r within equations and mathematical formulas.
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Note:
if the problem you need to solve does not provide the length of the beam, you will have to refer to one of the other sections of the article. In this case you will have to use the diameter or the area to be able to trace the length of the circumference.
Step 2. Draw the "diameter" of the circle
Extends the segment indicating the radius so that it passes through the center and reaches the opposite end of the circle. In other words, you have drawn a second ray. These two rays joined together represent the "diameter" of the circle, which is normally indicated by the letter d. At this point you will also have understood why you can calculate the diameter of a circle starting from the radius and vice versa, since the first measures exactly twice the second, i.e. d = 2r.
Step 3. Understand the meaning of the constant π ("pi")
The symbol π, which refers to the Greek letter pi, does not represent a magic number that randomly works for geometry problems; in reality the π was "discovered" precisely by measuring the circumference of the circles. If you try to measure the circumference of any circle (for example using a meter) and divide it by the length of the diameter, you will always get the same result, ie the value of the constant pi. This is a very special number because it cannot be reported in the form of a simple fraction or a decimal number, since it has an infinite number of digits. However, as a general rule, its rounded shape is used, which we all know to be equal to 3, 14.
The value of the constant π stored in calculators also does not use the real number, although it uses one that comes very close to it
Step 4. Take note of the mathematical definition of the constant π
As explained above, the constant π indicates the relationship between the circumference of a circle and its diameter. Placing this definition in mathematical terms you will get the following equation: π = C / d. Since you know that the diameter of any circle is equal to twice the radius, i.e. 2r, the formula just obtained can be rewritten as follows: π = C / 2r.
C is the variable that indicates the "circumference" of a circle
Step 5. Solve the equation obtained in the previous step based on C to find the circumference of a circle
Since your aim is to calculate the length of the circumference of a circle, you have to solve the given equation based on the variable C. Multiplying both sides of the equation by 2r you will get π x 2r = (C / 2r) x 2r, which simplifying is like writing 2πr = C.
- The left side of the formula can also be indicated in the form π2r; however it is correct. Numbers are usually given before variables in formulas so that equations are easier to read and understand. This step does not change the final result of the equation.
- In mathematical equations it is always possible to multiply both sides by the same value and obtain an equivalent equation.
Step 6. Replace the variables of the formula with real numbers and perform calculations to find the value of C
Now that you know that the circumference of a circle can be calculated using the formula 2πr = C, refer to the original text of your geometry problem to find the value of r (i.e. the radius of the circle you are studying). Replace the constant π with the value 3, 14 or use a scientific calculator that is equipped with the "π" key to get a more precise result. Solve the expression "2πr" using the numbers you found (3, 14 and the length of the radius). The result you will get will be equal to the circumference of the circle in question.
- For example, if the radius of the circle you are looking at is 2 units, you will get 2πr = 2 x (3, 14) x (2 units) = 12, 56 units. In this example, the circumference will be 12.56 units.
- By solving the same example problem using a scientific calculator with the "π" key, you will get a more precise result: 2 x π x 2 units = 12, 56637. However, if your professor has not given you different instructions, you can round the result obtained at 12, 57 units.
Method 2 of 4: Calculate the Circumference Using the Diameter
Step 1. Understand what "diameter" means
Place the tip of a pencil on a sheet of paper where you have previously drawn a circle. Align the tip with the circumference of the latter. Now draw a line that, passing through the center of the circle, reaches the opposite point of the circumference. The segment you have just drawn represents the "diameter" of the circle in question, which is normally indicated with the variable d within math and geometry problems.
- The line you drew must pass exactly through the center of the circle, otherwise it will not represent its diameter.
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Note:
if the problem you need to solve does not provide the length of the diameter, you will have to refer to one of the other sections of the article to be able to trace the length of the circumference.
Step 2. Understand the meaning of the following equation d = 2r
The "radius" of a circle, usually indicated by the variable r, represents the distance that separates the center from any point on the circumference. Since the diameter is the segment that joins two opposite points of the circumference passing through the center, it is easy to guess that its length is equal to twice the radius. In other words, the following equation is always true: d = 2r. This means that, within an equation or formula, you can always substitute the variable d with 2r or viceversa.
In this case you will use the variable d and not the shape 2r, as the problem you will face will give you the length of the diameter d and not that of the ray. However, it is very important to understand the meaning of this step, so that you don't get confused if your professor or math book refers to the diameter. d with the value 2r.
Step 3. Understand the meaning of the constant π ("pi")
The symbol π, which refers to the Greek letter pi, does not represent a magic number that randomly works for geometry problems. In reality the π was "discovered" precisely by measuring the circumference of the circles. If you try to measure the circumference of any circle (for example using a meter) and divide it by the length of the diameter, you will always get the same result, ie the value of the constant pi. This is a very special number because it cannot be reported in the form of a simple fraction or a decimal number, since it has an infinite number of digits. However, as a general rule, we use its rounded shape which we all know to be equal to 3, 14.
The value of the constant π stored in calculators also does not use the real number, although it uses one that comes very close to it
Step 4. Take note of the mathematical definition of the constant π
As explained above, the constant π indicates the relationship between the circumference of a circle and its diameter. Placing this definition in mathematical terms you will get the following equation: π = C / d.
Step 5. Solve the equation given in the previous step, based on the variable C, to calculate the circumference
Since you want to calculate the length of the circumference of a circle, you will need to modify the formula under consideration so that the variable C is isolated in a member of the equation. To do this, multiply both sides of the formula by d:
- π x d = (C / d) x d;
- πd = C.
Step 6. Replace the variables of the formula with real numbers and perform calculations to find the value of C
Refer to the original text of your problem to find out the diameter value d and replace it inside the equation you got in the previous step. Replace the constant π with the value 3, 14 or use a scientific calculator that is equipped with the "π" key to get a more precise result. Multiply the values of π and d to obtain the value of C, the length of the circumference of the circle in question.
- For example, if the diameter of the circle you are looking at is 6 units, you will get 2πd = (3, 14) x (6 units) = 18, 84 units. In this example, the circumference will be 18.84 units.
- By solving the same example problem using a scientific calculator with a "π" key, you will get a more precise result: π x 6 units = 18.84956. However, if your professor has not given you different instructions, you can round up the result. at 18, 85 units.
Method 3 of 4: Calculate the Circumference Using Area
Step 1. Understand how the area of a circle is calculated
In most cases, the area (TO) of a circle. Normally you simply need to measure the radius (r) and then go back to the corresponding area using the following mathematical formula: A = πr2. The mathematical proof of the correctness of this formula is a bit complicated, but if you are interested you can get more information by reading this article.
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Note:
if the problem you need to solve does not provide the value of the area, you will have to refer to one of the other sections of the article to be able to trace the length of the circumference.
Step 2. Find out the formula for calculating the circumference of a circle
The circumference (C.) of a circle is the set of points equidistant from its center that delimit its area. Normally you can calculate it using the formula C = 2πr. However, since in this case you do not know directly the value of the radius (r), you will have to spend some time calculating its value.
Step 3. Go back to the formula that will allow you to calculate the radius of a circle from its area
Since the area of a circle is defined by the formula A = πr2, you can go back to the inverse formula by solving the equation based on the variable r. If the steps below seem too complex to you, try starting with simpler algebra problems or deepen your knowledge of algebra.
- A = πr2;
- A / π = πr2 / π = r2;
- √ (A / π) = √ (r2) = r;
- r = √ (A / π).
Step 4. Modify the initial formula to calculate the circumference using the equation you got in the previous step
When you face any equation, for example r = √ (A / π), know that you can replace a member with a corresponding shape. Use this technique to correctly modify the initial circumference formula C = 2πr. In this case you don't know the value of the variable "r" directly, but you know the value of the area, "A". Replace the variable "r" with the formula you got in the previous step, so you can do the calculations:
- C = 2πr;
- C = 2π (√ (A / π)).
Step 5. Replace the variables of the formula with the known values, in order to find the circumference
Use the area value given to you in the problem text and do the calculations to get the final result. For example, if the area (TO) of the circle in question is equal to 15 square units, solve the following calculation 2π (√ (15 / π)) using a calculator. Remember to also enter the round brackets in the formula, otherwise the result will not be correct.
The result you get from the example problem will be 13.72937. However, if your teacher didn't give you different instructions you can round the result to 13, 73 square units.
Method 4 of 4: Measure the Circumference of a Real Circle
Step 1. Use this method if you need to physically measure real circular objects
Remember that it is also possible to trace the circumference of objects in the real world, not just those described in the math and geometry problems. Try measuring the circumference of a wheel on your bicycle, a pizza or a coin.
Step 2. Get a piece of string or thread and a ruler
The rope must be long enough to be wrapped around the circumference of the object. In addition, it will also need to be very flexible so that it can be wrapped tightly around the object. At this point you need a tool with which to measure, for example a tape measure or a ruler. Taking the measurement will be easier if the ruler or tape measure is longer than the piece of string to be measured.
Step 3. Wrap the string around the object only once
Start by placing one end of the string on one side of the object to be measured. At this point, wrap it all around the circumference, making sure it is as taut as possible. If you have to measure a coin or a very thin object, you may not be able to properly pull the string or wire around the circumference. Place the object to be measured on a flat surface, then wrap the string around the base trying to stretch it as much as possible.
Be careful not to overlap the ends of the string or thread. You will only need to wrap the object once, otherwise the measurement will be skewed. At the end of this step, you should have a single loop of string that should not be double in any section
Step 4. Mark or cut the string
Find the point where the rope circle closes, i.e. return to the starting point. Now mark the point under examination with a felt-tip pen or pen or use a pair of scissors to cut the section of string that perfectly describes the circumference of the object to be measured.
Step 5. Now unfold the string and measure its length using a ruler or tape measure
If you have chosen to use a marker, you will need to measure the piece of string from the starting point to the mark you made. This is the piece of string that completely wrapped the circumference of the object and that will give you the answer you are looking for. The length of the section of rope under examination is equivalent to the circumference of the object.
