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Calculating the radius of a circle is a fundamental concept in mathematics. The radius is a crucial measurement that helps determine various properties of a circle, such as its area, circumference, and diameter. Understanding how to calculate the radius is essential not only in geometry but also in various scientific and engineering fields. This topic explores the step-by-step process of calculating the radius of a circle using different methods, including measurements, formulas, and mathematical equations. Whether you are a student, professional, or simply have an interest in mathematics, mastering how to calculate the radius of a circle is a valuable skill that can be applied in numerous practical scenarios. In this guide, we will delve into the basic principles behind circle radius calculations and provide clear examples to enhance your understanding and application of this essential concept.
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.
This article has been viewed 315,867 times.
The radius of a circle is the distance from the center of the circle to any point on the circumference of the circle. [1] XResearch 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=2π(r){displaystyle C=2pi (r)} ) or area ( A=π(r2){displaystyle A=pi (r^{2})} ) of the circle, you can still find the radius of the circle using formulas and split variables r{displaystyle r} .
Steps
Find the radius when the circumference of the circle is known
![Image titled Calculate the Radius of a Circle Step 4](https://www.wikihow.com/images_en/thumb/2/2f/Calculate-the-Radius-of-a-Circle-Step-4-Version-6.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-4-Version-6.jpg)
, in there OLD{displaystyle C} is the circumference, and r{displaystyle r} is the radius. [2] XResearch Source
- Symbol pi{displaystyle pi} (“pi”) is a special number, approximately 3.14. You can use this value (3,14) in calculations or use the symbol pi{displaystyle pi} on the computer.
![Image titled Calculate the Radius of a Circle Step 5](https://www.wikihow.com/images_en/thumb/2/21/Calculate-the-Radius-of-a-Circle-Step-5-Version-6.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-5-Version-6.jpg)
For example
OLD=2πr{displaystyle C=2pi r}
OLD2π=2πr2π{displaystyle {frac {C}{2pi }}={frac {2pi r}{2pi }}}
OLD2π=r{displaystyle {frac {C}{2pi }}=r}
r=OLD2π{displaystyle r={frac {C}{2pi }}}
![Image titled Calculate the Radius of a Circle Step 6](https://www.wikihow.com/images_en/thumb/4/48/Calculate-the-Radius-of-a-Circle-Step-6-Version-6.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-6-Version-6.jpg)
For example
If the circumference of the circle is 15 cm, we will have the formula: r=152π{displaystyle r={frac {15}{2pi }}} cm
![Image titled Calculate the Radius of a Circle Step 8](https://www.wikihow.com/images_en/thumb/8/8a/Calculate-the-Radius-of-a-Circle-Step-8-Version-4.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-8-Version-4.jpg)
For example
r=152π={displaystyle r={frac {15}{2pi }}=} about 7.52∗3,14={displaystyle {frac {7.5}{2*3.14}}=} approximately 2.39 cm
Find the radius when the area of the circle is known
![Image titled Calculate the Radius of a Circle Step 9](https://www.wikihow.com/images_en/thumb/3/31/Calculate-the-Radius-of-a-Circle-Step-9-Version-4.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-9-Version-4.jpg)
, in there A{displaystyle A} is the area of the circle, and r{displaystyle r} is the radius. [3] XResearch Sources
![Image titled Calculate the Radius of a Circle Step 10](https://www.wikihow.com/images_en/thumb/f/f3/Calculate-the-Radius-of-a-Circle-Step-10-Version-4.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-10-Version-4.jpg)
For example
Divide both sides by π{displaystyle pi } :
A=πr2{displaystyle A=pi r^{2}}
Aπ=r2{displaystyle {frac {A}{pi }}=r^{2}}
Take the square root of both sides:
Aπ=r{displaystyle {sqrt {frac {A}{pi }}}=r}
r=Aπ{displaystyle r={sqrt {frac {A}{pi }}}}
![Image titled Calculate the Radius of a Circle Step 11](https://www.wikihow.com/images_en/thumb/8/89/Calculate-the-Radius-of-a-Circle-Step-11-Version-4.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-11-Version-4.jpg)
For example
If the area of the circle is 21 square centimeters, this formula would be:r=21π{displaystyle r={sqrt {frac {21}{pi }}}}
![Image titled Calculate the Radius of a Circle Step 12](https://www.wikihow.com/images_en/thumb/9/93/Calculate-the-Radius-of-a-Circle-Step-12-Version-3.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-12-Version-3.jpg)
For example
If 3.14 is used instead of π{displaystyle pi } , we have the calculation:
r=213,14{displaystyle r={sqrt {frac {21}{3,14}}}}
r=6,69{displaystyle r={sqrt {6,69}}}
If the calculator allows entering the entire formula in a row, we will have a more accurate answer.
![Image titled Calculate the Radius of a Circle Step 13](https://www.wikihow.com/images_en/thumb/d/da/Calculate-the-Radius-of-a-Circle-Step-13-Version-3.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-13-Version-3.jpg)
, 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} . 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 csquare meter2=csquare meter{displaystyle {sqrt {cm^{2}}}=cm} .
Find the radius when the diameter of the circle is known
![Image titled Calculate the Radius of a Circle Step 1](https://www.wikihow.com/images_en/thumb/6/6b/Calculate-the-Radius-of-a-Circle-Step-1-Version-6.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-1-Version-6.jpg)
, touch both opposite points on the circle. [4] XResearch 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.
![Image titled Calculate the Radius of a Circle Step 3](https://www.wikihow.com/images_en/thumb/e/eb/Calculate-the-Radius-of-a-Circle-Step-3-Version-5.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-3-Version-5.jpg)
[5] XResearch 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=d2{displaystyle r={frac {d}{2}}} .
Calculate the radius when knowing the area and the angle at the center of the fan shape
![Image titled Calculate the Radius of a Circle Step 14](https://www.wikihow.com/images_en/thumb/7/7b/Calculate-the-Radius-of-a-Circle-Step-14-Version-3.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-14-Version-3.jpg)
, in there ASector{displaystyle A_{sector}} is the fan-shaped area, θ{displaystyle theta } is the angle at the center of the fan in degrees, and r{displaystyle r} is the radius of the circle. [6] XResearch Source
![Image titled Calculate the Radius of a Circle Step 15](https://www.wikihow.com/images_en/thumb/1/14/Calculate-the-Radius-of-a-Circle-Step-15-Version-3.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-15-Version-3.jpg)
We will substitute the value of the fan area for the variable ASector{displaystyle A_{sector}} and the central angle for the variable θ{displaystyle 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=120360(π)(r2){displaystyle 50={frac {120}{360}}(pi )(r^{2})} .
![Image titled Calculate the Radius of a Circle Step 16](https://www.wikihow.com/images_en/thumb/a/a1/Calculate-the-Radius-of-a-Circle-Step-16-Version-3.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-16-Version-3.jpg)
For example
120360=first3{displaystyle {frac {120}{360}}={frac {1}{3}}} , that is, the fan shape is equal to first3{displaystyle {frac {1}{3}}} circle.
We will have the following equation: 50=first3(π)(r2){displaystyle 50={frac {1}{3}}(pi )(r^{2})}
![Image titled Calculate the Radius of a Circle Step 17](https://www.wikihow.com/images_en/thumb/0/01/Calculate-the-Radius-of-a-Circle-Step-17-Version-3.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-17-Version-3.jpg)
For example
50=first3(π)(r2){displaystyle 50={frac {1}{3}}(pi )(r^{2})}
50first3=first3(π)(r2)first3{displaystyle {frac {50}{frac {1}{3}}}={frac {{frac {1}{3}}(pi )(r^{2})}{frac {1}{3}} }}
150=(π)(r2){displaystyle 150=(pi )(r^{2})}
![Image titled Calculate the Radius of a Circle Step 18](https://www.wikihow.com/images_en/thumb/e/e6/Calculate-the-Radius-of-a-Circle-Step-18-Version-3.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-18-Version-3.jpg)
For example
150=(π)(r2){displaystyle 150=(pi )(r^{2})}
150π=(π)(r2)π{displaystyle {frac {150}{pi }}={frac {(pi )(r^{2})}{pi }}}
47.7=r2{displaystyle 47.7=r^{2}}
![Image titled Calculate the Radius of a Circle Step 19](https://www.wikihow.com/images_en/thumb/f/f5/Calculate-the-Radius-of-a-Circle-Step-19-Version-3.jpg/v4-460px-Calculate-the-Radius-of-a-Circle-Step-19-Version-3.jpg)
For example
47,7=r2{displaystyle 47.7=r^{2}}
47,7=r2{displaystyle {sqrt {47,7}}={sqrt {r^{2}}}}
6,91=r{displaystyle 6.91=r}
Thus, the radius of the circle will be about 6.91 cm.
Advice
- Number pi{displaystyle pi} actually in a circle. If we measure the circumference C and diameter d of the circle very precisely, then the calculation OLD÷d{displaystyle Cdiv d} will return the number pi{displaystyle pi} .
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.
This article has been viewed 315,867 times.
The radius of a circle is the distance from the center of the circle to any point on the circumference of the circle. [1] XResearch 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=2π(r){displaystyle C=2pi (r)} ) or area ( A=π(r2){displaystyle A=pi (r^{2})} ) of the circle, you can still find the radius of the circle using formulas and split variables r{displaystyle r} .
In conclusion, calculating the radius of a circle is a relatively simple process that requires only a few steps. By utilizing the formula r = √(A/π), where r represents the radius and A represents the area of the circle, one can determine the length of the radius accurately. Additionally, by using the circumference formula C = 2πr, it is possible to calculate the radius if the circumference is known. Another method involves measuring the distance between the center of the circle and any point on its circumference. This can be achieved using a ruler or measuring tape. The radius is essential in various mathematical and geometric calculations, as well as in practical applications such as construction, architecture, and engineering. By understanding the concept of radius calculation and its significance, individuals can effectively apply it in different contexts to solve problems and find solutions.
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