Circle Perimeter (Circumference) Calculator
Calculate the perimeter (circumference) of a circle from radius, diameter, or area. Also find arc length, sector perimeter, chord length, and area with unit conversions.
Circle Theorems Calculator & Explorer
Explore and compute results for major circle theorems: inscribed angle, central angle, tangent-chord, secant-secant, tangent-tangent, power of a point, and Thales' theorem.
All Circle Theorems Reference
| Theorem | Formula | Description |
|---|---|---|
| Inscribed Angle | θ = arc / 2 | An inscribed angle is half its intercepted arc. |
| Central Angle | θ_central = 2 × θ_inscribed | A central angle is twice the inscribed angle subtending the same arc. |
| Tangent-Chord | θ = arc / 2 | Angle between tangent and chord equals half the intercepted arc. |
| Secant-Secant | θ = (far arc − near arc) / 2 | Angle formed by two secants from an external point. |
| Tangent-Tangent | θ = (major arc − minor arc) / 2 | Angle between two tangent lines from an external point. |
| Power of a Point (Internal) | PA × PB = PC × PD | Two chords intersecting inside a circle. |
| Power of a Point (External) | PA × PB = PC × PD | Two secants from an external point. |
| Thales' Theorem | Angle in semicircle = 90° | Any angle inscribed in a semicircle is a right angle. |
| Cyclic Quadrilateral | Opp. angles sum to 180° | Opposite angles of a quadrilateral inscribed in a circle are supplementary. |
| Equal Chords | Equal chords ⟹ equal arcs | Chords of equal length subtend equal arcs. |
Planning notes, formulas, and examples
About the Circle Theorems Calculator & Explorer
Circle theorems form the backbone of Euclidean circle geometry. They describe precise relationships between angles, arcs, chords, tangents, and secants, and they appear throughout high-school and college mathematics, standardized tests, and real engineering problems.
This interactive calculator and explorer covers the most important circle theorems from a single chooser. Select a theorem from the dropdown — <strong>Inscribed Angle</strong>, <strong>Central Angle</strong>, <strong>Tangent-Chord</strong>, <strong>Secant-Secant</strong>, <strong>Tangent-Tangent</strong>, <strong>Power of a Point</strong> (internal and external), and <strong>Thales' Theorem</strong> — and the inputs switch to match that case. Enter the relevant measurements and see the derived angle, arc fraction, or segment relationship tied directly to the theorem you selected.
Eight preset buttons demonstrate each theorem with quick-load values, so you can explore without manual entry. A comprehensive reference table lists every major circle theorem with its formula and a plain-English description, making this an ideal revision aid for exams. Visual comparison bars let you see how angles and values relate at a glance.
Whether you are preparing for the SAT, ACT, or a college geometry course, this page helps you verify solutions, build intuition, and understand the elegant connections among circle properties.
When This Page Helps
Use this when you want to check the exact theorem behind a circle-geometry problem instead of memorizing several disconnected formulas. It is useful for exam prep and proof work because each mode ties the chosen theorem to the corresponding angles, arcs, and secant or tangent relationships.
How to Use the Inputs
- Choose a circle theorem from the dropdown selector.
- Enter the required values in the input fields (angles in degrees, lengths in any consistent unit).
- Read the computed results in the output cards — each card includes an explanation.
- Use preset buttons to load example values for each theorem.
- Compare values visually using the bar chart below the outputs.
- Scroll down to the reference table to review all circle theorems and their formulas.
Formula used
Inscribed Angle = Arc/2. Central = 2 × Inscribed. Tangent-Chord = Arc/2. Secant-Secant θ = |far arc − near arc|/2. Power of a Point: PA × PB = PC × PD. Thales': Angle in semicircle = 90°.Example Calculation
Result: For theorem=inscribed, v1=80, the tool returns the solved circle theorems outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in circle theorems formulas and reports derived values, checks, and classifications automatically.
Tips & Best Practices
- The inscribed angle theorem is the most commonly tested circle theorem on standardized exams.
- Power of a Point works for both intersecting chords (internal) and intersecting secants (external).
- Thales' theorem is a special case of the inscribed angle theorem where the arc is a semicircle.
- Two inscribed angles intercepting the same arc are always equal, regardless of where on the circle they sit.
- A tangent line is always perpendicular to the radius drawn to the point of tangency.
When To Use This Calculator
Explore and compute results for major circle theorems: inscribed angle, central angle, tangent-chord, secant-secant, tangent-tangent, power of a point, and Thales Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.
How To Check The Result
Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.
Practical Notes
Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.
Sources & Methodology
Last updated:
Frequently Asked Questions
It states that an inscribed angle (vertex on the circle) is exactly half the central angle (or intercepted arc) that subtends the same arc.
When two chords intersect inside a circle, the products of their segments are equal: PA × PB = PC × PD. The same relationship holds for secants from an external point.
Thales' theorem says that any angle inscribed in a semicircle is a right angle (90°). The hypotenuse of the resulting right triangle is the diameter.
The Tangent-Tangent and Secant-Secant options cover these cases. A secant-tangent angle uses the same half-difference formula with one arc set to zero for the tangent.
A quadrilateral inscribed in a circle. Its opposite angles always sum to 180°.
Yes — in surveying (bearing calculations), optics (lens curvature), CAD/CAM machining (arc fitting), architecture (arch design), and satellite orbit geometry.
More in this topic
Circumcenter of a Triangle Calculator
Find the circumcenter, circumradius, centroid, incenter, orthocenter, perpendicular bisector equations, area, and perimeter of a triangle from 3 vertex coordinates.
Arc Length Calculator
Calculate arc length, sector area, chord length, and arc-to-chord ratio from radius and central angle. Includes preset common angles, a properties table, and an arc diagram.