How to Find the Missing Radius or Central Angle of a Sector

Most geometry problems ask you to find the area of a sector. However, in real-world applications like architectural design, metalworking, or landscaping, you may already know the area you need to cover. In these cases, you must work backward using algebra to find the required radius or central angle.

Finding the Central Angle from Sector Area

If you know the radius ($r$) and the sector area ($A$), you can isolate the central angle ($\theta$). The formula differs based on whether you want your final angle in degrees or radians.

Formula for Central Angle in Degrees:

$$\theta = \frac{360 \times A}{\pi r^2}$$

Example: You have a sector with an area of $50 \text{ in}^2$ and a radius of $6$ in. What is the angle in degrees?
$$\theta = \frac{360 \times 50}{\pi (36)^2}$$
$$\theta = \frac{18000}{113.097}$$
$$\theta \approx 159.15^\circ$$

Formula for Central Angle in Radians:

$$\theta = \frac{2A}{r^2}$$

Example: You have a sector with an area of $24 \text{ cm}^2$ and a radius of $4$ cm. What is the angle in radians?
$$\theta = \frac{2 \times 24}{16}$$
$$\theta = \frac{48}{16}$$
$$\theta = 3 \text{ radians}$$

Finding the Radius from Sector Area

If you know the sector's area ($A$) and its central angle ($\theta$), you can isolate the radius. Because the radius is squared in the standard formula, finding it requires calculating a square root.

Formula for Radius (When angle is in degrees):

$$r = \sqrt{\frac{360 \times A}{\pi \times \theta}}$$

Formula for Radius (When angle is in radians):

$$r = \sqrt{\frac{2A}{\theta}}$$

Automating Reverse Calculations

Working backward requires isolating variables, managing square roots, and keeping strict track of Pi ($\pi$) to avoid rounding errors. The easiest way to verify your algebra is to use a tool designed for reverse calculations. You can select "Find Central Angle" or "Find Radius" directly inside our sector of a circle calculator to instantly solve for these missing geometric variables.