Radius of Circle Calculator Using Degrees Area of A Sector

This calculator helps you find the radius of a circle when you know the area of a sector and the angle in degrees. Understanding how to calculate the radius from a sector's area is useful in geometry, engineering, and design applications where you need to work with circular measurements.

How to Use This Calculator

To use this radius of circle calculator:

Enter the area of the sector in square units (e.g., cm², m², in²).
Enter the angle of the sector in degrees (between 0 and 360).
Click the "Calculate" button to see the radius.
The result will appear below the calculator with the calculated radius.

The calculator will show you the exact radius based on your inputs. You can also see a visual representation of the sector if you prefer a graphical understanding.

Formula Explained

The relationship between the area of a sector, the angle, and the radius is given by the formula:

Area of Sector = (θ/360) × π × r²

Where:

θ is the angle of the sector in degrees
π is the mathematical constant pi (approximately 3.14159)
r is the radius of the circle

To find the radius, we rearrange the formula to solve for r:

r = √[(Area of Sector × 360) / (θ × π)]

This formula allows you to calculate the radius when you know the sector area and angle. The calculator uses this exact formula to provide accurate results.

Worked Example

Let's say you have a sector with an area of 15 cm² and an angle of 60 degrees. Here's how to calculate the radius:

Plug the values into the formula: r = √[(15 × 360) / (60 × π)]
Calculate the numerator: 15 × 360 = 5400
Calculate the denominator: 60 × π ≈ 60 × 3.14159 ≈ 188.4955
Divide numerator by denominator: 5400 / 188.4955 ≈ 28.6479
Take the square root: √28.6479 ≈ 5.352 cm

So, the radius of the circle is approximately 5.352 cm. You can verify this result using our calculator by entering 15 for the area and 60 for the angle.

Frequently Asked Questions

What is the difference between a sector and a circle?

A sector is a pie-shaped part of a circle enclosed by two radii and an arc. A circle is the complete set of all points in a plane that are at a given distance (the radius) from a given point (the center).

Can I use this calculator for full circles?

Yes, if you enter 360 degrees as the angle, the calculator will give you the radius of the full circle. The area of the full circle would be πr², which matches the formula when θ=360.

What if I enter an angle greater than 360 degrees?

The calculator will still work, but angles greater than 360 degrees will represent multiple full rotations around the circle. The result will be the radius of the circle, and the sector area will be calculated for the equivalent angle within one full rotation (θ mod 360).