Inscribed Angle Calculator
Calculate the inscribed angle from the central angle using the inscribed angle theorem. Free online geometry calculator with instant results and formula explanations.
About This Calculator
The Inscribed Angle Calculator helps you find the inscribed angle of a circle when you know the central angle. An inscribed angle is formed by two chords that intersect on the circle's circumference, while the central angle has its vertex at the circle's center. Both angles subtend the same arc, and the inscribed angle theorem states that the inscribed angle is always exactly half of the central angle.
This calculator is ideal for students learning circle theorems, teachers preparing geometry lessons, engineers working with circular structures, and anyone studying geometry. The relationship θi = θc / 2 is one of the most important circle theorems and appears in many geometry problems involving cyclic quadrilaterals, arc length calculations, and chord properties.
How to use: Enter the central angle in degrees and click Calculate. The calculator instantly shows the inscribed angle. The input accepts any positive angle value up to 360°.
Examples: A central angle of 90° produces an inscribed angle of 45°. A central angle of 180° (a diameter) gives 90° — the well-known "angle in a semicircle" theorem that the angle inscribed by a diameter is always a right angle.
Formula
The inscribed angle theorem: θi = θc / 2
Where θi is the inscribed angle and θc is the central angle. Both angles subtend the same arc. This theorem also implies that all inscribed angles subtending the same arc are equal, regardless of where the vertex lies on the circumference.
Regional Notes
This calculator uses degrees, the standard unit for angle measurement worldwide. The inscribed angle theorem is a universal geometric principle taught in curricula across India (CBSE, ICSE), the United States (Common Core), and the United Kingdom (GCSE, A-Level). No region-specific adjustments are needed.
Frequently Asked Questions
What is the inscribed angle theorem?
The inscribed angle theorem states that an inscribed angle is half the measure of the central angle that subtends the same arc. Mathematically, θi = θc / 2, where θc is the central angle and θi is the inscribed angle.
How do you find the inscribed angle from the central angle?
To find the inscribed angle, simply divide the central angle by 2. For example, if the central angle is 60°, the inscribed angle is 30°. If the central angle is 120°, the inscribed angle is 60°.
What is the relationship between inscribed and central angles?
The inscribed angle is always exactly half of the central angle when both angles subtend the same arc. This relationship holds for any circle and any arc length, and changing the position of the inscribed angle vertex along the circumference does not change its measure.
What is the inscribed angle of a diameter?
The angle inscribed by the two endpoints of a diameter is always 90° (a right angle). This is because the central angle subtended by the diameter is 180°, and the inscribed angle is half of that, giving 90°.
Can the inscribed angle be greater than 90°?
Yes, the inscribed angle can be greater than 90° when the central angle exceeds 180°. However, inscribed angles are typically measured between 0° and 180°, corresponding to central angles between 0° and 360°.
What if I enter a central angle of 0°?
A central angle of 0° means the arc has zero length, so the inscribed angle is also 0°. The calculator handles this case correctly and returns 0° for both the central and inscribed angles.
Why is understanding inscribed angles important?
Inscribed angles are fundamental in geometry and trigonometry. They appear in circle theorems, cyclic quadrilateral problems, arc length calculations, and real-world applications like engineering design, architecture, and astronomy.
Is the Inscribed Angle Calculator free to use?
Yes, the Inscribed Angle Calculator on Calculy is completely free to use with no registration required. You can calculate as many angles as you need and share results via URL.