Sector Area Calculator – Circle Sector with Radius & Angle
A sector is the “slice” of a circle bounded by two radii and the arc between them.
Its main ingredients are the radius \(r\), the central angle \(\theta\), and the sector area \(A\).
1) Sector area in radians
When \(\theta\) is measured in radians, the sector area is:
\[
A=\frac{1}{2}r^2\theta.
\]
Reason: a full circle corresponds to \(\theta=2\pi\), and the fraction of the circle is \(\theta/(2\pi)\).
2) Sector area in degrees
If \(\theta\) is measured in degrees, use the fraction of a full turn:
\[
A=\frac{\theta}{360}\cdot \pi r^2.
\]
3) Converting degrees ↔ radians
\[
\theta_{\text{rad}}=\theta_{\deg}\cdot\frac{\pi}{180},
\qquad
\theta_{\deg}=\theta_{\text{rad}}\cdot\frac{180}{\pi}.
\]
In most formulas below, treat \(\theta\) as radians unless stated otherwise.
4) Solving for the angle from area and radius
Start from \(A=\tfrac{1}{2}r^2\theta\) and solve for \(\theta\) (in radians):
\[
\theta=\frac{2A}{r^2}.
\]
If you want degrees, convert:
\[
\theta_{\deg}=\left(\frac{2A}{r^2}\right)\cdot\frac{180}{\pi}.
\]
5) Solving for the radius from area and angle
If the sector area \(A\) and central angle \(\theta\) are known, solve for \(r\).
Using the radians form:
\[
A=\frac{1}{2}r^2\theta
\;\;\Longrightarrow\;\;
r^2=\frac{2A}{\theta}
\;\;\Longrightarrow\;\;
r=\sqrt{\frac{2A}{\theta}}.
\]
If your angle is given in degrees, convert it to radians first:
\(\theta_{\text{rad}}=\theta_{\deg}\cdot\pi/180\).
6) Useful related formulas
-
Circle area:
\[
A_{\text{circle}}=\pi r^2.
\]
-
Fraction of the circle:
\[
\frac{A}{\pi r^2}=\frac{\theta}{2\pi}.
\]
-
Arc length of the sector:
\[
s=r\theta \quad (\theta \text{ in radians}).
\]
-
Chord length between arc endpoints:
\[
c=2r\sin\left(\frac{\theta}{2}\right).
\]
-
Segment area (sector minus triangle):
\[
A_{\text{segment}}=\frac{1}{2}r^2\left(\theta-\sin\theta\right).
\]
Example (radius + angle → area)
Given \(r=5\) and \(\theta=90^\circ\):
\[
A=\frac{90}{360}\cdot \pi\cdot 5^2
=\frac{1}{4}\cdot 25\pi
=\frac{25\pi}{4}
\approx 19.63.
\]
Example (area + angle → radius)
Given \(A=\frac{25\pi}{4}\) and \(\theta=\frac{\pi}{2}\) (radians):
\[
r=\sqrt{\frac{2A}{\theta}}
=\sqrt{\frac{2\cdot \frac{25\pi}{4}}{\frac{\pi}{2}}}
=\sqrt{25}=5.
\]
