The Circle Theorem Calculator is a quick-check workspace for matching a circle diagram to a theorem and checking the resulting angle or segment. This guide specifically explains this calculator: its ten choices under Choose a theorem, its changing numeric fields, its exact result labels, its formulas, and its supporting controls.
The calculator page is intentionally short and easy to scan. It is meant for quick revision before solving a geometry question, not as a replacement for identifying the points, chords, tangents, and segments in your own diagram. Use the page for the check, then use this guide to choose the correct theorem, enter the right values, interpret the output, and verify the working.
What does the Circle Theorem Calculator do?
The calculator presents a drop-down menu labelled Choose a theorem. The menu contains Angle at the centre, Inscribed angle theorem, Angle in a semicircle, Angles in the same segment, Cyclic quadrilateral, Tangent and radius, Alternate segment, Perpendicular from centre to chord, Intersecting chords, and Tangent-secant.
After a choice is made, the page displays a plain-language explanation and the Click-to-check workspace. Depending on the theorem, the workspace creates zero, one, two, or three number fields. The fields are HTML number inputs with a minimum of zero and support decimal values. There is no unit selector on this calculator. Angle fields are labelled with (°); chord and secant fields are plain lengths, so use one consistent unit in a problem and carry that unit through your own work.
What are the calculator's theorem choices and outputs?
| Theorem choice | Exact input label or labels | Formula or relationship | Exact result label |
|---|---|---|---|
| Angle at the centre | Circumference angle (°) | Centre angle = 2 × circumference angle | Centre angle = value° |
| Inscribed angle theorem | Centre angle (°) | Inscribed angle = centre angle ÷ 2 | Inscribed angle = value° |
| Angle in a semicircle | No input field | An angle subtended by a diameter is 90° | Angle = 90° |
| Angles in the same segment | Angle A (°) | Angle A = Angle B | The second angle = value° |
| Cyclic quadrilateral | Angle A (°) | A + C = 180° | Opposite angle = value° |
| Tangent and radius | No input field | Radius ⊥ tangent, so the angle is 90° | Angle = 90° |
| Alternate segment | Tangent-chord angle (°) | Tangent-chord angle = alternate angle | Alternate angle = value° |
| Perpendicular from centre to chord | Full chord length | Half chord = full chord ÷ 2 | Each half chord = value |
| Intersecting chords | a, b, c | a × b = c × d | Missing segment d = value |
| Tangent-secant | External secant, Whole secant | t2 = external × whole secant | Tangent = value |
What do the circle-theorem terms mean?
Angle at the centre
The centre angle has its vertex at the centre of the circle, while the circumference angle stands on the same arc with its vertex on the circle. The calculator doubles the supplied Circumference angle (°) and returns Centre angle = ...°. The relationship is θcentre = 2θcircumference.
Inscribed angle theorem
An inscribed angle is an angle whose vertex lies on the circumference. When it and a centre angle stand on the same arc, the inscribed angle is half the supplied Centre angle (°). The result is labelled Inscribed angle = ...°, using θinscribed = θcentre ÷ 2.
Angle in a semicircle
A diameter subtends a right angle at any point on the circumference. Selecting Angle in a semicircle creates no numeric input field because the result is fixed. The calculator returns Angle = 90°.
Angles in the same segment
Two angles standing on the same chord and in the same segment have equal measures. Enter Angle A (°); the tool copies that value into The second angle = ...°. In symbols, ∠A = ∠B.
Cyclic quadrilateral
A cyclic quadrilateral has all four vertices on the circumference. Its opposite angles are supplementary, meaning their sum is 180°. Enter Angle A (°); the calculator reports Opposite angle = ...° by applying A + C = 180°, or C = 180° − A.
Tangent and radius
A tangent touches a circle at one point. The radius drawn to that point of contact is perpendicular to the tangent. Like the semicircle theorem, this choice needs no number field and returns Angle = 90°.
Alternate segment
The angle between a tangent and a chord equals the angle in the alternate segment of the circle. Enter Tangent-chord angle (°); the output is Alternate angle = ...°. The calculator uses equality rather than a subtraction or doubling step.
Perpendicular from centre to chord
A perpendicular from the centre to a chord bisects that chord. Enter the complete measurement in Full chord length. The result, Each half chord = ..., is one half of the supplied length. The point where the perpendicular meets the chord is its midpoint.
Intersecting chords
When two chords intersect inside a circle, the product of the two segments on one chord equals the product of the two segments on the other. The fields are labelled exactly a, b, and c. The calculator solves for Missing segment d = ... using a × b = c × d, so d = (a × b) ÷ c.
Tangent-secant
From one outside point, a tangent and a secant satisfy the tangent-secant theorem. Enter the two fields labelled External secant and Whole secant. The output Tangent = ... is found from t2 = external secant × whole secant, followed by t = √(external secant × whole secant).
Does this calculator use π?
This calculator does not use π in any of its ten checks. The formulas here concern angle relationships, perpendicularity, equality, segment products, and a square root rather than circumference, area, arcs, or sectors. Therefore, do not add 3.14159 to an input or expect a unit conversion.
The relevant fixed constants are 90° for a right angle, 180° for the sum of opposite cyclic-quadrilateral angles, the factor 2 for the centre-angle relationship, and √ for recovering a tangent from a squared length. The degree symbol belongs to angle values. The chord and secant calculations preserve whatever common length unit you use outside the page.
How to use the Circle Theorem Calculator
- Open the Circle Theorem Calculator.
- Choose one theorem under Choose a theorem.
- Read the explanation shown above the workspace and identify the exact field labels that appear.
- Enter only positive numeric values where the selected theorem requires them. For the two fixed results, Angle in a semicircle and Tangent and radius, there are no input fields.
- Click Click to check example.
- Read the bold result, then use Click to visualize solution or Click to show steps if you want a diagram or a short calculation check.
The page does not provide a separate copy button. If you need to transfer a result, copy it only after checking the theorem name, the input labels, the numerical values, the degree symbol or length context, and the formula shown in the steps.
Worked example using every theorem option
The following checks use every option in the live menu and retain the calculator's exact field and result labels. The displayed numeric formatter can remove unnecessary trailing zeroes, so the examples below show the expected mathematical result rather than a forced number of decimal places.
| Choose a theorem | Value entered in the exact field | Click | Expected calculator output |
|---|---|---|---|
| Angle at the centre | Circumference angle (°) = 35 | Click to check example | Centre angle = 70°, because 2 × 35° = 70° |
| Inscribed angle theorem | Centre angle (°) = 100 | Click to check example | Inscribed angle = 50°, because 100° ÷ 2 = 50° |
| Angle in a semicircle | No field is shown | Click to check example | Angle = 90° |
| Angles in the same segment | Angle A (°) = 42 | Click to check example | The second angle = 42° |
| Cyclic quadrilateral | Angle A (°) = 68 | Click to check example | Opposite angle = 112°, because 180° − 68° = 112° |
| Tangent and radius | No field is shown | Click to check example | Angle = 90° |
| Alternate segment | Tangent-chord angle (°) = 37 | Click to check example | Alternate angle = 37° |
| Perpendicular from centre to chord | Full chord length = 18 | Click to check example | Each half chord = 9, because 18 ÷ 2 = 9 |
| Intersecting chords | a = 6, b = 10, c = 15 | Click to check example | Missing segment d = 4, because (6 × 10) ÷ 15 = 4 |
| Tangent-secant | External secant = 9; Whole secant = 16 | Click to check example | Tangent = 12, because √(9 × 16) = √144 = 12 |
How the exact formulas are checked
| Selected theorem | Calculator working | What to verify |
|---|---|---|
| Angle at the centre | Double the circumference angle: 2 × input = centre angle | The two angles stand on the same arc. |
| Inscribed angle theorem | Centre angle ÷ 2 = inscribed angle | The circumference angle and centre angle intercept the same arc. |
| Angle in a semicircle | Diameter → right angle → 90° | The line forming the diameter passes through the centre. |
| Angles in the same segment | Angle B = Angle A | The angles stand on the same chord and are in the same segment. |
| Cyclic quadrilateral | 180° − Angle A = opposite angle | All four vertices lie on the same circle. |
| Tangent and radius | Radius ⊥ tangent → 90° | The radius ends at the point where the tangent touches. |
| Alternate segment | Tangent-chord angle = angle in alternate segment | The selected angle really uses the tangent and the named chord. |
| Perpendicular from centre to chord | Full chord ÷ 2 = each half chord | The perpendicular begins at the centre and meets the chord at its midpoint. |
| Intersecting chords | d = (a × b) ÷ c | The chords cross inside the circle and the segment labels match the equation. |
| Tangent-secant | t = √(external secant × whole secant) | The whole secant is the complete outside-to-far-intersection distance, not only its inside part. |
What do the validation messages mean?
The calculator checks that required inputs are positive. For Angle at the centre, Angles in the same segment, and Alternate segment, the entered angle must be greater than zero. For Inscribed angle theorem, the centre angle must be between 0° and 360°. For Cyclic quadrilateral, the entered angle must be between 0° and 180°. Perpendicular from centre to chord asks for a positive full chord length, while Intersecting chords requires all three positive segment values and Tangent-secant requires both positive lengths.
If a required field is blank, zero, or invalid for the selected theorem, the page shows an error instead of a result. Correct the value and click Click to check example again. The two fixed theorems do not need a numeric entry.
How do visualization, steps, clear, and history work?
After a successful check, Click to visualize solution becomes available. It opens an in-page diagram of the selected relationship. The diagram may show a centre, radius, tangent, chord, intersecting chords, or an outside point depending on the theorem, followed by a short explanatory note. It is a visual confirmation of the relationship, not a scale drawing of your particular diagram.
Select Click to show steps to reveal the calculator's short working. The same control changes to Hide steps while the working is visible. The steps show doubling, halving, subtraction from 180°, chord bisection, a segment-product rearrangement, or the square-root calculation as appropriate.
Click Clear to reset the current theorem explanation, fields, result, steps, and visualization. Clearing the workspace does not mean that the saved recent checks have been deleted. To inspect saved checks, click Click to view history. When the panel opens, that control changes to Hide history and the panel includes Recent checks, Clear history, and the note that checks are saved only on this device.
The history list keeps the most recent ten successful checks and records the theorem choice plus its result. Click Clear history only when you intentionally want to remove those local entries. The page has no Click to copy result or Click to copy solution control, so copying must be done manually after verification.
Where Do We Use Circle-Theorem Results in Real Life?
These outputs are mainly used to check geometry in design, surveying, manufacturing, construction, and technical drawing. Each result answers a different question, so the theorem choice matters more than simply obtaining a number.
| Calculator output | Distinct practical use |
|---|---|
| Centre angle or inscribed angle | A CAD or mechanical-design workflow can use the matching central and rim angles to locate holes, spokes, markers, or repeated features around a circular part. |
| 90° from a semicircle or a radius and tangent | A builder, machinist, or surveyor can check a right-angle condition where a diameter, a circular boundary, or a tangent contact establishes a perpendicular reference. |
| Equal angle in the same segment | Technical drawings can confirm that two sight lines or construction lines subtend the same chord and therefore preserve the same angle at different points on the circle. |
| Opposite angle in a cyclic quadrilateral | Surveying and structural layouts can test four-point circular arrangements, such as marked positions around a circular boundary, by checking that opposite angles total 180°. |
| Alternate angle | Road, pipe, and machine-part drawings can verify the angle made when a straight tangent-like edge meets a chord-like line on a circular feature. |
| Each half chord | Manufacturing a round plate, window, or bracket may require the midpoint of a chord so that a perpendicular centreline can be marked accurately. |
| Missing segment d | Inspection drawings for circular frames or drilled parts can recover an unmeasured segment when two intersecting chords provide the other three measurements. |
| Tangent | Wheel, pulley, cam, and belt layouts can use the tangent length associated with an external secant relationship to check clearance or contact geometry. |
When is this calculator not the right tool?
This calculator checks named circle theorems; it does not calculate a circle's area, circumference, arc length, chord length from an angle, sector area, or complete coordinate equation. Use the Basic Circle Calculator for standard radius, diameter, circumference, and area relationships. Use the Circle Arc Calculator for a partial curved distance and the Circle Chord Calculator when the task is to calculate a chord.
For a sector, segment, or ring, use the Sector, Segment and Ring Calculator. For a circle equation, use the Circle Equation Calculator. For a circle defined by three coordinate points, use the Circle from 3 Points Calculator. Those tools address measurement or coordinate problems that the theorem workspace intentionally does not attempt.
Final verification workflow
- Identify the geometric facts in the diagram: same arc, diameter, same chord, cyclic quadrilateral, tangent, perpendicular-to-chord line, intersecting chords, or external tangent and secant.
- Choose the matching option under Choose a theorem; do not choose an option merely because its output name sounds similar to the question.
- Check the exact field labels. Enter a circumference angle only for Angle at the centre, a centre angle only for Inscribed angle theorem, and the three or two segment values in the order shown for the product theorems.
- Keep degrees for angle inputs and keep the same length unit for every chord or secant value. Remember that this page has no unit menu.
- Click Click to check example and compare the result with the appropriate formula, including 90°, 180°, halving, doubling, a product division, or a square root.
- Use Click to show steps and, when useful, Click to visualize solution. Confirm that the picture's relationship matches your original diagram.
- Before copying the answer manually, check the theorem name, input values, result label, units or degree symbol, and any rounding shown by the calculator.
- Use Clear for a new problem. Use Click to view history only when comparing recent checks, and remember that the history is local to the device.
Open the Circle Theorem Calculator for the quick check, then return to this guide whenever you need to explain why the selected theorem and output are correct.
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