Circle Radius Calculator: Find Radius from Diameter & Area

Radius of a circle

The radius of a circle is the distance from the circle's center to any point on its edge. It can be found from the diameter, the circumference, or the area, using three simple formulas. This calculator computes the radius from whichever of those three values is known.

What is a radius?

A circle is a curved line made of every point that sits the same distance from one fixed point, called the center. That fixed distance is the radius. All radii of a single circle are equal in length. The radius sets the size of the circle: a larger radius makes a bigger circle.

Three other measurements relate directly to the radius:

• Diameter ($d$): a straight line from one edge of the circle to the other, passing through the center. It is always twice the radius.
• Circumference ($C$): the distance around the outside of the circle.
• Area ($A$): the amount of space enclosed by the circle.

How to calculate the radius of a circle

The formula used depends on which measurement is already known.

From the diameter

$$r = \frac{d}{2}$$

The diameter is twice the radius, so dividing it by 2 gives the radius back.

From the circumference

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

This comes from the circumference formula $C = 2\pi r$, solved for $r$. The Greek letter $\pi$ (pi) is a constant, approximately 3.14159.

From the area

$$r = \sqrt{\frac{A}{\pi}}$$

This comes from the area formula $A = \pi r^2$, solved for $r$.

Worked examples

Example 1: radius from diameter

A circle has a diameter of 10 units.

$$r = \frac{10}{2} = 5 \text{ units}$$

Example 2: radius from circumference

A circle has a circumference of 31.4159 units.

$$r = \frac{31.4159}{2\pi} \approx \frac{31.4159}{6.2832} \approx 5 \text{ units}$$

Example 3: radius from area

A circle has an area of 78.5398 square units.

$$r = \sqrt{\frac{78.5398}{\pi}} \approx \sqrt{25} = 5 \text{ units}$$

Using the calculator

Choose which value is known — diameter, circumference, or area — and enter a number greater than zero. The calculator returns the radius rounded to four decimal places. Entering zero, a negative number, or text instead of a number produces an error message, because a circle cannot have a diameter, circumference, or area of zero or less.

The radius is given in the same unit as the input. Entering a diameter in centimeters returns a radius in centimeters; entering an area in square meters returns a radius in meters.

Why the radius matters

The radius is the one measurement from which every other property of a circle can be derived. It appears in the formulas for diameter, circumference, area, and the volume and surface area of a sphere. Engineers use it to size pipes, gears, and wheels. Architects use it to design arches and domes. Astronomers use it to describe the size of planets, stars, and orbits.

A short history

Circles were studied by ancient Egyptian and Babylonian mathematicians, who worked out early approximations of the ratio between a circle's circumference and its diameter. Around 300 BCE, the Greek mathematician Euclid gave a formal definition of a circle and its radius in his book Elements. Later, Archimedes developed a method for estimating $\pi$ by comparing a circle to polygons drawn inside and outside it. Mathematicians in China, India, and elsewhere refined the value of $\pi$ over the following centuries, arriving at the digits used in calculations today.

Frequently asked questions

What is the radius of a circle?

The radius is the distance from the center of a circle to any point on its edge. It is half the length of the diameter.

How do you find the radius from the diameter?

Divide the diameter by 2: $r = d/2$. A circle with a diameter of 10 units has a radius of 5 units.

How do you find the radius from the circumference?

Divide the circumference by $2\pi$: $r = C/(2\pi)$. A circle with a circumference of about 31.42 units has a radius of about 5 units.

How do you find the radius from the area?

Divide the area by $\pi$, then take the square root: $r = \sqrt{A/\pi}$. A circle with an area of about 78.54 square units has a radius of about 5 units.

What is the difference between radius and diameter?

The diameter is a straight line through the center connecting two points on the circle. It is exactly twice the radius: $d = 2r$.

Can the radius be negative or zero?

No. A real circle must have a radius greater than zero. The calculator shows an error if the diameter, circumference, or area entered is zero, negative, or not a number.

References

1. Circle — Wikipedia
2. Pi — Wikipedia