how to find the radius of a circle with the circumference

Download Article

Download Article

The radius of a circle is the distance from the center of the circle to any point on its circumference.^[1] The easiest way to find the radius is by dividing the diameter in half. If you don't know the diameter but you know other measurements, such as the circle's circumference ( ${\displaystyle C=2\pi r}$ ) or area ( ${\displaystyle A=\pi r^{2}}$ ), you can still find the radius by using the formulas and isolating the ${\displaystyle r}$ variable.

1. 1

   Write down the circumference formula. The formula is

   ${\displaystyle C=2\pi r}$

   , where ${\displaystyle C}$ equals the circle's circumference, and ${\displaystyle r}$ equals its radii^[2]

2. 2

   Solve for r. Use algebra to change the circumference formula until r (radius) is alone on one side of the equation:

   Example
   ${\displaystyle C=2\pi r}$
   ${\displaystyle {\frac {C}{2\pi }}={\frac {2\pi r}{2\pi }}}$
   ${\displaystyle {\frac {C}{2\pi }}=r}$
   ${\displaystyle r={\frac {C}{2\pi }}}$

3. 3

   Plug the circumference into the formula. Whenever a math problem tells you the circumference C of a circle, you can use this equation to find the radius r. Replace C in the equation with the circumference of the circle in your problem:

   Example
   If the circumference is 15 centimeters, your formula will look like this: ${\displaystyle r={\frac {15}{2\pi }}}$ centimeters

4. 4

   Round to a decimal answer. Enter your result in a calculator with the ${\displaystyle \pi }$ button and round the result. If you don't have a calculator, calculate it by hand, using 3.14 as a close estimate for ${\displaystyle \pi }$ .

   Example
   ${\displaystyle r={\frac {15}{2\pi }}=}$ about ${\displaystyle {\frac {7.5}{2*3.14}}=}$ approximately 2.39 centimeters

1. 1

   Set up the formula for the area of a circle. The formula is

   ${\displaystyle A=\pi r^{2}}$

   , where ${\displaystyle A}$ equals the area of the circle, and ${\displaystyle r}$ equals the radius.^[3]

2. 2

   Solve for the radius. Use algebra to get the radius r alone on one side of the equation:

   Example
   Divide both sides by ${\displaystyle \pi }$ :
   ${\displaystyle A=\pi r^{2}}$
   ${\displaystyle {\frac {A}{\pi }}=r^{2}}$
   Take the square root of both sides:
   ${\displaystyle {\sqrt {\frac {A}{\pi }}}=r}$
   ${\displaystyle r={\sqrt {\frac {A}{\pi }}}}$

3. 3

   Plug the area into the formula. Use this formula to find the radius when the problem tells you the area of the circle. Substitute the area of the circle for the variable ${\displaystyle A}$ .

   Example
   If the area of the circle is 21 square centimeters, the formula will look like this: ${\displaystyle r={\sqrt {\frac {21}{\pi }}}}$

4. 4

   Divide the area by $$ π {\displaystyle \pi } . Begin solving the problem by simplifying the portion under the square root ( ${\displaystyle {\frac {A}{\pi }})}$ . Use a calculator with a ${\displaystyle \pi }$ key if possible. If you don't have a calculator, use 3.14 as an estimate for ${\displaystyle \pi }$ .

   Example
   If using 3.14 for ${\displaystyle \pi }$ , you would calculate:
   ${\displaystyle r={\sqrt {\frac {21}{3.14}}}}$
   ${\displaystyle r={\sqrt {6.69}}}$
   If your calculator allows you to enter the whole formula on one line, that will give you a more accurate answer.

5. 5

   Take the square root.

   You will likely need a calculator to do this

   , because the number will be a decimal. This value will give you the radius of the circle.

   Example
   ${\displaystyle r={\sqrt {6.69}}=2.59}$ . So, the radius of a circle with an area of 21 square centimeters is about 2.59 centimeters.
   Areas always use square units (like square centimeters), but the radius always uses units of length (like centimeters). If you keep track of units in this problem, you'll notice that ${\displaystyle {\sqrt {cm^{2}}}=cm}$ .

1. 1

   Check the problem for a diameter. If the problem tells you the diameter of the circle, it's easy to find the radius. If you are working with an actual circle,

   measure the diameter by placing a ruler so its edge passes straight through the circle's center

   , touching the circle on both sides.^[4]

   • If you're not sure where the circle center is, put the ruler down across your best guess. Hold the zero mark of the ruler steady against the circle, and slowly move the other end back and forth around the circle's edge. The highest measurement you can find is the diameter.
   • For example, you might have a circle with a diameter of 4 centimeters.

2. 2

   Divide the diameter by two. A circle's

   radius is always half the length of its diameter.

   ^[5]

   • For example, if the diameter is 4 cm, the radius equals 4 cm ÷ 2 = 2 cm.
   • In math formulas, the radius is r and the diameter is d. You might see this step in your textbook as ${\displaystyle r={\frac {d}{2}}}$ .

1. 1

   Set up the formula for the area of a sector. The formula is

   ${\displaystyle A_{sector}={\frac {\theta }{360}}(\pi )(r^{2})}$

   , where ${\displaystyle A_{sector}}$ equals the area of the sector, ${\displaystyle \theta }$ equals the central angle of the sector in degrees, and ${\displaystyle r}$ equals the radius of the circle.^[6]

2. 2

   Plug the sector's area and central angle into the formula. This information should be given to you.

   Make sure you have the area of the sector, not the area for the circle.

   Substitute the area for the variable ${\displaystyle A_{sector}}$ and the angle for the variable ${\displaystyle \theta }$ .

   Example
   If the area of the sector is 50 square centimeters, and the central angle is 120 degrees, you would set up the formula like this:
   ${\displaystyle 50={\frac {120}{360}}(\pi )(r^{2})}$ .

3. 3

   Divide the central angle by 360. This will tell you what fraction of the entire circle the sector represents.

   Example
   ${\displaystyle {\frac {120}{360}}={\frac {1}{3}}}$ . This means that the sector is ${\displaystyle {\frac {1}{3}}}$ of the circle.
   Your equation should now look like this: ${\displaystyle 50={\frac {1}{3}}(\pi )(r^{2})}$

4. 4

   Isolate $$ ( π ) ( r 2 ) {\displaystyle (\pi )(r^{2})} . To do this, divide both sides of the equation by the fraction or decimal you just calculated.

   Example
   ${\displaystyle 50={\frac {1}{3}}(\pi )(r^{2})}$
   ${\displaystyle {\frac {50}{\frac {1}{3}}}={\frac {{\frac {1}{3}}(\pi )(r^{2})}{\frac {1}{3}}}}$
   ${\displaystyle 150=(\pi )(r^{2})}$

5. 5

   Divide both sides of the equation by $$ π {\displaystyle \pi } . This will isolate the ${\displaystyle r}$ variable. For a more precise result, use a calculator. You can also round ${\displaystyle \pi }$ to 3.14.

   Example
   ${\displaystyle 150=(\pi )(r^{2})}$
   ${\displaystyle {\frac {150}{\pi }}={\frac {(\pi )(r^{2})}{\pi }}}$
   ${\displaystyle 47.7=r^{2}}$

6. 6

   Take the square root of both sides. This will give you the radius of the circle.

   Example
   ${\displaystyle 47.7=r^{2}}$
   ${\displaystyle {\sqrt {47.7}}={\sqrt {r^{2}}}}$
   ${\displaystyle 6.91=r}$
   So, the radius of the circle is about 6.91 centimeters.

Add New Question

• Question

  How do I find the radius of a circle when I know the chord length?

  

  It is possible to have quite a few circles, all with different radii, in which one could draw a chord of a given, fixed length. Hence, the chord length by itself cannot determine the radius of the circle.

• Question

  How do I find the radius of a circle when I know the arc length and the central angle?

  

  Divide the central angle into 360°. Multiply the resulting number by the arc length. That gives you the circumference of the circle. Divide the circumference by pi. That's the diameter. Half of the diameter is the radius of the circle.

• Question

  How do I calculate the radius of a circle when no other values are known?

  

  Technically you can't "calculate" the radius in such a situation. However, it is possible, by construction, to locate the center of such a circle, and then, simply by physically measuring, determine the radius. To do the construction, draw any two chords and construct their perpendicular bisectors; their point of intersection is the center of the circle. Then draw in any radius and measure it with a ruler. Not technically a "calculation."

• Question

  How do I find the circumference and area of a circle with a diameter of 11?

  

  STEP 1 : radius = D/2. 11/2 = 5.5 STEP 2 : circumference = 2 x pi x r = (2)(pi)( 5.5) = 11 x pi = 34.54 STEP 3 : area = pi x r^2 = pi (5.5 * 5.5 ) = 30.25 x pi = 94.99

• Question

  How do I find the measure of an arc without a radius?

  

  Figure out the ratio of the length of the arc to the circumference and set it equal to the ratio of the measure of the arc (shown with a variable) and the measure of the entire circle (360 degrees). For example: If the circumference of the circle is 4 and the length of the arc is 1, the proportion would be 4/1 = 360/x and x would equal 90.

• Question

  How do you calculate the radius of a circle when only the area is given?

  

  Take the square root of the area of the circle and divide by pi.

• Question

  How do I calculate the area of a circle using the radius?

  

  The area of a circle is equal to pi times the radius squared.

• Question

  How do I calculate the radius of a circle if the circumference is 1.76?

  

  The circumference is equal to the radius multiplied by 2pi. Given the circumference, divide the circumference by 2pi and that's your radius. In this case, your radius should be around .28.

• Question

  How do I find the circumference of a circle with a 4.75 cm radius?

  

  By using the formula C = 2* pi * r =(2)(3.141)(4.75) = 29.84 cm

• Question

  How do I figure out the radius with just an arc length?

  

  You can't do it without additional information.

Show more answers

Ask a Question

200 characters left

Include your email address to get a message when this question is answered.

Submit

Thanks for submitting a tip for review!

About This Article

Article SummaryX

To calculate the radius of a circle by using the circumference, take the circumference of the circle and divide it by 2 times π. For a circle with a circumference of 15, you would divide 15 by 2 times 3.14 and round the decimal point to your answer of approximately 2.39. Be sure to include the units in your answer. To learn more, such as how to calculate the radius with the area or diameter, keep reading the article!

Did this summary help you?

Thanks to all authors for creating a page that has been read 3,057,291 times.

Did this article help you?

how to find the radius of a circle with the circumference

Source: https://www.wikihow.com/Calculate-the-Radius-of-a-Circle

Posted by: garciasciales.blogspot.com

Share this post