How to Calculate the Radius of a Circle

X

This article was co-written by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math teacher at City University of San Francisco and previously worked in the math department of Saint Louis University. She has taught math at the elementary, middle, high, and college levels. She holds a master’s degree in education from Saint Louis University, majoring in management and supervision in education.

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The radius of a circle is the distance from the center of the circle to any point on the circumference of the circle. ^{[1] X Research Source} The easiest way to calculate the radius of a circle is to divide the diameter of the circle in half. If you don’t know the diameter of the circle but know other measurements, such as circumference ( $OLD$ $=$ $2$ $π$ $($ $r$ $)$ ${displaystyle C=2pi (r)}$ $How to Calculate the Radius of a Circle$ ) or area ( $A$ $=$ $π$ $($ $r$ $2$ $)$ ${displaystyle A=pi (r^{2})}$ $A=pi (r^{{2}})$ ) of the circle, you can still find the radius of the circle using formulas and split variables $r$ ${displaystyle r}$ $r$ .

Table of Contents

Find the radius when the circumference of the circle is known

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Write the formula for the circumference of a circle. This formula is

OLD

=

2

π

r

{displaystyle C=2pi r}

C=2pi r

, in there $OLD$ ${displaystyle C}$ $OLD$ is the circumference, and $r$ ${displaystyle r}$ $r$ is the radius. ^{[2] X Research Source}

Symbol $p$ $i$ ${displaystyle pi}$ $pi$ (“pi”) is a special number, approximately 3.14. You can use this value (3,14) in calculations or use the symbol $p$ $i$ ${displaystyle pi}$ $pi$ on the computer.

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Calculate r (radius). Use algebra to convert the formula for the circumference of a circle until only r (radius) remains on one side of the equation:

For example
$OLD$ $=$ $2$ $π$ $r$ ${displaystyle C=2pi r}$ $C=2pi r$
$OLD$ $2$ $π$ $=$ $2$ $π$ $r$ $2$ $π$ ${displaystyle {frac {C}{2pi }}={frac {2pi r}{2pi }}}$ ${frac {C}{2pi }}={frac {2pi r}{2pi }}$
$OLD$ $2$ $π$ $=$ $r$ ${displaystyle {frac {C}{2pi }}=r}$ ${frac {C}{2pi }}=r$
$r$ $=$ $OLD$ $2$ $π$ ${displaystyle r={frac {C}{2pi }}}$ $r={frac {C}{2pi }}$

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Substitute the perimeter value into the formula. When the problem tells you the value of C for the circumference of a circle, you can use this equation to find the radius r . We will substitute the value C of the circumference of the circle in the math problem into the equation:

For example
If the circumference of the circle is 15 cm, we will have the formula: $r$ $=$ $15$ $2$ $π$ ${displaystyle r={frac {15}{2pi }}}$ $r={frac {15}{2pi }}$ cm

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Round to decimal. Enter the results into the calculator with the button

π

{displaystyle pi }

pi

and round the number. If you don’t have a calculator, you can do the math by hand, using 3.14 as an approximate value of

π

{displaystyle pi }

pi

.

For example
$r$ $=$ $15$ $2$ $π$ $=$ ${displaystyle r={frac {15}{2pi }}=}$ $r={frac {15}{2pi }}=$ about $7.5$ $2$ $*$ $3$ $,$ $14$ $=$ ${displaystyle {frac {7.5}{2*3.14}}=}$ ${frac {7.5}{2*3.14}}=$ approximately 2.39 cm

Find the radius when the area of the circle is known

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Write down the formula for calculating the area of a circle. This formula is

A

=

π

r

2

{displaystyle A=pi r^{2}}

A=pi r^{{2}}

, in there $A$ ${displaystyle A}$ $A$ is the area of the circle, and $r$ ${displaystyle r}$ $r$ is the radius. ^{[3] X Research Sources}

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Solve the equation to find the radius. Use algebra to return r to one side of the equation:

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

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Substitute the area value into the formula. Use this formula to find the radius if the problem gives the area value of the circle. We will substitute the value of the area of the circle for the variable

A

{displaystyle A}

A

.

For example
If the area of the circle is 21 square centimeters, this formula would be: $r$ $=$ $21$ $π$ ${displaystyle r={sqrt {frac {21}{pi }}}}$ $r={sqrt {{frac {21}{pi }}}}$

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Divide the area by the number $π$ ${displaystyle pi }$ $pi$ . Start by reducing the part below the square root sign (

A

π

)

{displaystyle {frac {A}{pi }})}

{frac {A}{pi }})

. Use the calculator with the . button

π

{displaystyle pi }

pi

if possible. If we don’t have a calculator, we’ll use 3.14 as the value of .

π

{displaystyle pi }

pi

.

For example
If 3.14 is used instead of $π$ ${displaystyle pi }$ $pi$ , we have the calculation:
$r$ $=$ $21$ $3$ $,$ $14$ ${displaystyle r={sqrt {frac {21}{3,14}}}}$ $r={sqrt {{frac {21}{3,14}}}}$
$r$ $=$ $6$ $,$ $69$ ${displaystyle r={sqrt {6,69}}}$ $r={sqrt {6,69}}$
If the calculator allows entering the entire formula in a row, we will have a more accurate answer.

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Calculate square root.

Maybe you need to use a calculator to do this calculation

, since this is a decimal. The result will be the radius of the circle.

For example
$r$ $=$ $6$ $,$ $69$ $=$ $2$ $,$ $59$ ${displaystyle r={sqrt {6.69}}=2.59}$ $r={sqrt {6.69}}=2.59$ . Thus, the radius of a circle with an area of 21 square centimeters is about 2.59 centimeters.
Area always uses square units (like square centimeters), but radius always uses length units (like centimeters). If you look at the units in this problem, you’ll notice $c$ $square meter$ $2$ $=$ $c$ $square meter$ ${displaystyle {sqrt {cm^{2}}}=cm}$ ${sqrt {cm^{{2}}}}=cm$ .

Find the radius when the diameter of the circle is known

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Find the diameter of the circle in the problem. The radius of a circle will be very easy to calculate if the problem is given the diameter data. If you are calculating on a specific circle,

You can measure the diameter by placing the ruler in the circle so that the edge of the ruler passes through the center of the circle

, touch both opposite points on the circle. ^{[4] X Research Sources}

If you are not sure where the center of the circle is, place the ruler across the circle according to your estimate. Keep the zero line on the ruler always close to the circle and slowly move the other end of the ruler around the circle. The largest measurement you will find will be the diameter measurement.
For example, your circle might have a diameter of 4 cm.

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Divide the diameter in half. Radius of the circle

is always half the length of the diameter.

^{[5] X Research Sources}

For example, if the circle’s diameter is 4 cm then its radius will be 4 cm ÷ 2 = 2 cm .
In the mathematical formula, the radius is denoted by r and the diameter is d . This formula in a textbook can be written as follows: $r$ $=$ $d$ $2$ ${displaystyle r={frac {d}{2}}}$ $r={frac {d}{2}}$ .

Calculate the radius when knowing the area and the angle at the center of the fan shape

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Write the formula for calculating the area of a fan. This formula is

A

S

e

c

t

o

r

=

θ

360

(

π

)

(

r

2

)

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

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

, in there $A$ $S$ $e$ $c$ $t$ $o$ $r$ ${displaystyle A_{sector}}$ $A_{{sector}}$ is the fan-shaped area, $θ$ ${displaystyle theta }$ $theta$ is the angle at the center of the fan in degrees, and $r$ ${displaystyle r}$ $r$ is the radius of the circle. ^{[6] X Research Source}

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Substitute the area and angle at the center of the fan into the formula. This information is in the title.

Remember that this is the area of the fan, not the circle.

We will substitute the value of the fan area for the variable $A$ $S$ $e$ $c$ $t$ $o$ $r$ ${displaystyle A_{sector}}$ $A_{{sector}}$ and the central angle for the variable $θ$ ${displaystyle theta }$ $theta$ .

For example
If the area of the fan is 50 square centimeters, and the angle at the center is 120 degrees, we have the following formula:
$50$ $=$ $120$ $360$ $($ $π$ $)$ $($ $r$ $2$ $)$ ${displaystyle 50={frac {120}{360}}(pi )(r^{2})}$ $50={frac {120}{360}}(pi )(r^{{2}})$ .

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Divide the central angle by 360. That way we will know how much of the circle the fan occupies.

For example
$120$ $360$ $=$ $first$ $3$ ${displaystyle {frac {120}{360}}={frac {1}{3}}}$ ${frac {120}{360}}={frac {1}{3}}$ , that is, the fan shape is equal to $first$ $3$ ${displaystyle {frac {1}{3}}}$ ${frac {1}{3}}$ circle.
We will have the following equation: $50$ $=$ $first$ $3$ $($ $π$ $)$ $($ $r$ $2$ $)$ ${displaystyle 50={frac {1}{3}}(pi )(r^{2})}$ $50={frac {1}{3}}(pi )(r^{{2}})$

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

For example
$50$ $=$ $first$ $3$ $($ $π$ $)$ $($ $r$ $2$ $)$ ${displaystyle 50={frac {1}{3}}(pi )(r^{2})}$ $50={frac {1}{3}}(pi )(r^{{2}})$
$50$ $first$ $3$ $=$ $first$ $3$ $($ $π$ $)$ $($ $r$ $2$ $)$ $first$ $3$ ${displaystyle {frac {50}{frac {1}{3}}}={frac {{frac {1}{3}}(pi )(r^{2})}{frac {1}{3}} }}$ ${frac {50}{{frac {1}{3}}}}={frac {{frac {1}{3}}(pi )(r^{{2}})}{{frac {1}{ 3}}}}$
$150$ $=$ $($ $π$ $)$ $($ $r$ $2$ $)$ ${displaystyle 150=(pi )(r^{2})}$ $150=(pi )(r^{{2}})$

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Divide both sides of the equation by the number $π$ ${displaystyle pi }$ $pi$ . This step will separate the variable

r

{displaystyle r}

r

. For more accurate results, you can use a calculator. You can also round numbers

π

{displaystyle pi }

pi

to 3.14.

For example
$150$ $=$ $($ $π$ $)$ $($ $r$ $2$ $)$ ${displaystyle 150=(pi )(r^{2})}$ $150=(pi )(r^{{2}})$
$150$ $π$ $=$ $($ $π$ $)$ $($ $r$ $2$ $)$ $π$ ${displaystyle {frac {150}{pi }}={frac {(pi )(r^{2})}{pi }}}$ ${frac {150}{pi }}={frac {(pi )(r^{{2}})}{pi }}$
$47.7$ $=$ $r$ $2$ ${displaystyle 47.7=r^{2}}$ $47.7=r^{{2}}$

Advice

Number $p$ $i$ ${displaystyle pi}$ $pi$ actually in a circle. If we measure the circumference C and diameter d of the circle very precisely, then the calculation $OLD$ $\div$ $d$ ${displaystyle Cdiv d}$ $Cdiv d$ will return the number $p$ $i$ ${displaystyle pi}$ $pi$ .

Steps

Find the radius when the circumference of the circle is known

Find the radius when the area of the circle is known

Find the radius when the diameter of the circle is known

Calculate the radius when knowing the area and the angle at the center of the fan shape

Advice

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