Radius of a circle calculator

What is the radius of a circle?

The radius of a circle is the distance from its center to any point on its edge. It is the most fundamental measurement of a circle: every other quantity — the diameter, the circumference, and the area — can be written in terms of the radius. Knowing the radius is like holding the key to the whole circle.

In practice you often measure something else first: the width across a wheel (its diameter), the length of a band wrapped around a tank (its circumference), or the painted surface of a round table (its area). This calculator works backwards from any of those, recovering the radius and then filling in the remaining quantities for you.

Diameter

The diameter $(d)$ stretches all the way across the circle through the center, so it is exactly twice the radius. Halving it gives the radius directly: $r = \frac{d}{2}$.

Circumference

The circumference $(C)$ is the distance around the circle, related to the radius by $C = 2\pi r$. Solving for the radius gives $r = \frac{C}{2\pi}$, where $\pi \approx 3.14159$.

Area

The area $(A)$ is the surface enclosed by the circle, given by $A = \pi r^2$. Rearranging for the radius gives $r = \sqrt{\frac{A}{\pi}}$.

Formulas

Each route to the radius follows from the basic circle relationships:

1. Radius from diameter:

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

2. Radius from circumference:

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

3. Radius from area:

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

Examples

Example 1: Radius from diameter

Suppose a circle has a diameter of 10 units. The radius is simply half the diameter:

$$r = \frac{d}{2} = \frac{10}{2} = 5$$

For reference, this circle has a circumference of $C = 2\pi r \approx 31.41593$ and an area of $A = \pi r^2 \approx 78.53982$.

Example 2: Radius from circumference

Now suppose only the circumference is known, $C = 31.41593$. Divide by $2\pi$:

$$r = \frac{C}{2\pi} = \frac{31.41593}{2 \times 3.14159} \approx 5$$

Example 3: Radius from area

Finally, suppose the area is $A = 78.53982$. Divide by $\pi$ and take the square root:

$$r = \sqrt{\frac{A}{\pi}} = \sqrt{\frac{78.53982}{3.14159}} = \sqrt{25} = 5$$

All three methods agree: the radius is 5.

Notes

• Half the diameter: When the diameter is known, no $\pi$ is involved — just divide by two.
• Units: The radius shares the same linear unit as the diameter and circumference (cm, m, in, …), while the area must be in the matching squared unit. Keep them consistent.
• Precision: More decimal places of $\pi$ produce a more precise radius; two or three places suffice for most everyday tasks.

Frequently asked questions

How do I find the radius if the diameter is 10?

Divide the diameter by two: $r = \frac{10}{2} = 5$.

How do I find the radius from the circumference?

Divide the circumference by $2\pi$. For $C = 31.41593$, the radius is $\frac{31.41593}{2 \times 3.14159} \approx 5$.

How do I find the radius from the area?

Use $r = \sqrt{A/\pi}$. For $A = 78.53982$, this gives $\sqrt{78.53982/3.14159} = \sqrt{25} = 5$.

What is the difference between radius and diameter?

The radius reaches from the center to the edge, while the diameter reaches all the way across through the center. The diameter is always exactly twice the radius. To go the other way and solve for the diameter, use the circle diameter calculator.

If the radius doubles, what happens to the area?

The area is proportional to the square of the radius, so doubling the radius quadruples the area. You can see this with the circle area calculator.

Why does the radius appear in so many circle formulas?

Because the radius is the defining measurement of a circle: the diameter, circumference, and area are all simple functions of it, which is why finding the radius effectively describes the entire circle.