Inscribed Angle Calculator

What is an Inscribed Angle?

An inscribed angle is an angle whose vertex lies directly on the circumference of a circle and whose sides (rays) contain chords of the circle. The arc that lies in the interior of the inscribed angle and has endpoints on the angle's rays is called the intercepted arc (or subtended arc).

Inscribed angles are foundational in Euclidean geometry, trigonometry, and computer graphics. With our Inscribed Angle Calculator, you can instantly find the inscribed angle $\psi$, the central angle $\theta$, the subtended arc length $s$, chord length $c$, sector area, and segment area from any given parameter.

The Inscribed Angle Theorem

The Inscribed Angle Theorem states that an angle inscribed in a circle is exactly half the measure of the central angle that subtends the same arc:

$$\psi = \frac{1}{2} \theta_{\text{central}}$$

Equivalently, the central angle is twice the inscribed angle:

$$\theta_{\text{central}} = 2 \psi$$

For further calculations on angles at the center of a circle, check out our Central Angle Calculator and explore broader geometric relationships with our Circle Theorems Calculator.

Key Geometric Corollaries

From the Inscribed Angle Theorem, several important geometric properties follow:

• Inscribed Angles on the Same Arc: Any two inscribed angles that subtend the exact same arc or congruent arcs are congruent ($\psi_1 = \psi_2$).
• Thales's Theorem (Angle in a Semicircle): If an inscribed angle subtends a diameter (a central angle of $180^\circ$), the inscribed angle is always a right angle ($90^\circ$).
• Opposite Angles of a Cyclic Quadrilateral: In any four-sided polygon inscribed in a circle, opposite interior angles are supplementary and sum to $180^\circ$.

Formulas for Arc Length, Chord Length, and Areas

Given a circle with radius $r$ and an inscribed angle $\psi$ (with corresponding central angle $\theta = 2\psi$):

• Subtended Arc Length ($s$): $s = r \cdot \theta_{\text{rad}} = r \cdot \left(\frac{2\psi \cdot \pi}{180}\right)$
• Subtended Chord Length ($c$): $c = 2r \sin(\psi)$ (or use our Chord Length Calculator)
• Circular Sector Area ($A_{\text{sector}}$): $A_{\text{sector}} = \frac{1}{2} r^2 \theta_{\text{rad}}$
• Circular Segment Area ($A_{\text{segment}}$): $A_{\text{segment}} = \frac{1}{2} r^2 (\theta_{\text{rad}} - \sin\theta_{\text{rad}})$
• Full Circle Properties: Find total area $\pi r^2$ and circumference $2\pi r$ using our Circle Calculator.

Step-by-Step Example

Suppose you have a circle of radius $r = 10\text{ cm}$ and an inscribed angle $\psi = 30^\circ$:

1. Find the central angle: $\theta = 2 \times 30^\circ = 60^\circ = \frac{\pi}{3}\text{ rad} \approx 1.0472\text{ rad}$.
2. Find the arc length: $s = r \cdot \theta_{\text{rad}} = 10 \times 1.0472 \approx 10.472\text{ cm}$.
3. Find the chord length: $c = 2 \times 10 \times \sin(30^\circ) = 20 \times 0.5 = 10\text{ cm}$.
4. Find the sector area: $A_{\text{sector}} = \frac{1}{2} \times 10^2 \times 1.0472 \approx 52.36\text{ cm}^2$.