Inscribed Angle Calculator
Calculate the inscribed angle from the central angle using the inscribed angle theorem: inscribed = central/2. An inscribed angle has its vertex on the circle and is always half the central angle subtending the same arc, fundamental to circle theorems.
This free online inscribed angle calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
The central angle in degrees
Results
Inscribed Angle
40 degrees
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Inscribed Angle Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Inscribed Angle Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
Formula Reference
Inscribed Angle Calculator Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
When to Use This Calculator
- •Use the Inscribed Angle Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
About This Calculator
The Inscribed Angle Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the inscribed angle from the central angle using the inscribed angle theorem: inscribed = central/2. An inscribed angle has its vertex on the circle and is always half the central angle subtending the same arc, fundamental to circle theorems. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.
About Inscribed Angle Calculator
The Inscribed Angle Calculator computes the inscribed angle from the central angle using the inscribed angle theorem. An inscribed angle has its vertex on the circle and its sides are chords of the circle. The theorem states that an inscribed angle is exactly half the central angle that subtends the same arc. This elegant result has numerous corollaries: all inscribed angles subtending the same arc are equal, an angle inscribed in a semicircle is 90 degrees (Thales' theorem), and opposite angles of a cyclic quadrilateral sum to 180 degrees. The inscribed angle theorem is used in circle geometry proofs, architecture, optics, and computer graphics.
The Math Behind It
Formula Reference
Inscribed Angle Theorem
inscribed angle = central angle / 2
Variables: Both angles subtend the same arc of the circle
Worked Examples
Example 1: Central Angle 80 degrees
A central angle of 80 degrees subtends an arc. Find the inscribed angle.
The inscribed angle is 40 degrees.
Example 2: Semicircle (Thales' Theorem)
The central angle is 180 degrees (a diameter). Find the inscribed angle.
The inscribed angle is 90 degrees.
Common Mistakes & Tips
- !Doubling instead of halving. The inscribed angle is HALF the central angle, not double. The central angle is the larger one.
- !Assuming the inscribed angle depends on where the vertex is placed on the circle. All inscribed angles subtending the same arc are equal, regardless of vertex position.
- !Applying the theorem to angles not inscribed in the circle. The vertex must be ON the circle, not at the center or outside.
Related Concepts
Used in These Calculators
Calculators that build on or apply the concepts from this page:
Frequently Asked Questions
Why is an angle inscribed in a semicircle always 90 degrees?
Because the central angle for a semicircle is 180 degrees (the diameter subtends a 180-degree arc). By the inscribed angle theorem, the inscribed angle is 180/2 = 90 degrees. This is Thales' theorem, one of the oldest known geometric results.
Can two different inscribed angles subtending the same arc be different?
No. All inscribed angles subtending the same arc are equal, regardless of where on the circle the vertex is placed. This is a direct consequence of the inscribed angle theorem.
What if the inscribed angle subtends a major arc instead of a minor arc?
If the inscribed angle subtends the major arc (greater than 180 degrees), the inscribed angle is still half the central angle of that major arc. The inscribed angle from the minor arc plus the inscribed angle from the major arc sum to 180 degrees.