Circle Theorems
Apply the inscribed angle theorem with this geometry calculator. Enter two angles to compute central, inscribed, sum, and difference for circle theorem proofs.
About This Calculator
The Circle Theorems Calculator is an interactive geometry tool that demonstrates the inscribed angle theorem and related angle relationships in circles. Students, teachers, and self-learners can use it to verify homework, prepare for exams, and build a visual understanding of how central and inscribed angles relate to each other.
The core formula implemented is the inscribed angle theorem: an inscribed angle is half the measure of the central angle that subtends the same arc. If the central angle is θ, the inscribed angle is θ/2. The calculator also computes the sum of both input angles and their absolute difference, giving a complete picture of the angle relationships being explored.
How to Use
Enter any two angle values in degrees into the First Angle and Second Angle fields, then click Calculate. The results panel shows the central angle (equal to your first input), the inscribed angle (half of the first angle), the sum of both input angles, and their difference. The inputs and results can be saved as a shareable URL for later reference or classroom discussion.
Key Circle Theorems at a Glance
- Inscribed Angle Theorem — An inscribed angle equals half the central angle subtending the same arc.
- Angle in a Semicircle — Any angle inscribed in a semicircle is a right angle (90°).
- Angles in the Same Segment — Angles subtended by the same chord on the same side of the chord are equal.
- Cyclic Quadrilateral — Opposite angles of a cyclic quadrilateral sum to 180°.
- Alternate Segment Theorem — The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment.
- Tangent-Radius Theorem — The radius drawn to the point of tangency is perpendicular to the tangent.
These theorems are foundational to Euclidean geometry and appear in curricula worldwide, including CBSE (India), GCSE (UK), and high school geometry (US). Understanding them is essential for solving problems involving circle properties, chord lengths, tangents, and angle chasing in competitive exams and standardized tests.
Frequently Asked Questions
What is the inscribed angle theorem?
The inscribed angle theorem states that an angle inscribed in a circle (formed by two chords from a point on the circumference) is half the measure of the central angle that subtends the same arc. For example, if the central angle is 60°, the inscribed angle that intercepts the same arc is 30°. This calculator applies this theorem by dividing the first angle by 2 to find the corresponding inscribed angle.
How do I use the Circle Theorems calculator?
Enter any two angle values in degrees into the input fields and click Calculate. The tool instantly shows the central angle, the inscribed angle (half the first angle), the sum of both angles, and their absolute difference. Use the result values to verify geometry homework, prepare for exams, or explore circle theorem relationships interactively.
What are the common circle theorems?
The six major circle theorems are: (1) Inscribed angle theorem — an inscribed angle is half the central angle on the same arc. (2) Angle in a semicircle — an angle inscribed in a semicircle is a right angle (90°). (3) Angles in the same segment are equal. (4) Opposite angles of a cyclic quadrilateral sum to 180°. (5) The alternate segment theorem. (6) Tangent-radius theorem — a radius meets a tangent at 90°. This calculator demonstrates theorem 1 directly.
What grade level is this calculator suitable for?
Circle theorems are typically introduced in grades 9–12 (GCSE and A-Level in the UK, high school geometry in the US and India). Students preparing for board exams (CBSE, ICSE, GCSE, SAT) or competitive tests can use this tool to check their manual calculations and build intuition about angle relationships in circles.
Is this calculator accurate for all angle values?
Yes. The calculator uses standard geometry formulas with double-precision arithmetic. Results are displayed to two decimal places. For the inscribed angle theorem, the inscribed angle is computed as exactly half the central angle. Edge cases such as zero or negative inputs are handled gracefully — the calculator requires both fields to have valid positive numbers before computing.
Can I share my calculation results?
Yes. After entering your values and calculating, the input parameters are saved to the page URL. You can copy the browser address and share it with classmates, teachers, or tutors. Anyone opening the link will see the same angle values and results automatically.