Mirror Equation Calculator
Solve the governing mirror equation relationship with editable SI inputs and scenario checks.
Mirror Equation Calculator inputs
Results and formula-based derivation
Mirror Equation Calculator: equations, variables, units and worked solution
The mirror equation calculator calculates Mirror Equation — image distance, Mirror magnification, Mirror Equation — image height, Radius of curvature from Mirror focal length, Object distance, Object height. It does not hide the arithmetic: the result panel shows the governing formula, canonical-unit conversion, numeric substitution, unrounded evaluation and final rounded answer for every output.
Variables and measurement units
| Symbol | Variable | Canonical unit | Minimum | Maximum |
|---|---|---|---|---|
focalLength | Mirror focal length | m | 1e-12 | 1000 |
objectDistance | Object distance | m | 1e-06 | 1000 |
objectHeight | Object height | m | 1e-06 | 1000 |
For the Mirror Equation Calculator, Mirror focal length, Object distance, and Object height are normalized to m, m, and m before 1/(1/focalLength-1/objectDistance) is evaluated. The unrounded value used for 1/(1/focalLength-1/objectDistance) and Mirror Equation — image distance is retained internally; formatting is applied only to the result cards.
Formula model
Spherical-mirror imaging uses 1/f = 1/do + 1/di; radius of curvature is twice the focal length.
| Output | Unit | Exact engine expression |
|---|---|---|
| Mirror Equation — image distance | m | 1 ÷ (1 ÷ focalLength-1 ÷ objectDistance) |
| Mirror magnification | x | -(1 ÷ (1 ÷ focalLength-1 ÷ objectDistance)) ÷ objectDistance |
| Mirror Equation — image height | m | objectHeight × -(1 ÷ (1 ÷ focalLength-1 ÷ objectDistance)) ÷ objectDistance |
| Radius of curvature | m | 2 × focalLength |
Formula-based worked derivation
- Mirror Equation — image distance
- Formula:
Mirror Equation — image distance = 1 ÷ (1 ÷ focalLength-1 ÷ objectDistance) - Default substitution:
Mirror Equation — image distance = 1 ÷ (1 ÷ (0.2)-1 ÷ (0.6)) - Unrounded evaluation:
0.3 m - Displayed answer: 0.3 m
- Formula:
- Mirror magnification
- Formula:
Mirror magnification = -(1 ÷ (1 ÷ focalLength-1 ÷ objectDistance)) ÷ objectDistance - Default substitution:
Mirror magnification = -(1 ÷ (1 ÷ (0.2)-1 ÷ (0.6))) ÷ (0.6) - Unrounded evaluation:
-0.5 x - Displayed answer: -0.5 x
- Formula:
- Mirror Equation — image height
- Formula:
Mirror Equation — image height = objectHeight × -(1 ÷ (1 ÷ focalLength-1 ÷ objectDistance)) ÷ objectDistance - Default substitution:
Mirror Equation — image height = (0.08) × -(1 ÷ (1 ÷ (0.2)-1 ÷ (0.6))) ÷ (0.6) - Unrounded evaluation:
-0.04 m - Displayed answer: -0.04 m
- Formula:
- Radius of curvature
- Formula:
Radius of curvature = 2 × focalLength - Default substitution:
Radius of curvature = 2 × (0.2) - Unrounded evaluation:
0.4 m - Displayed answer: 0.4 m
- Formula:
In the Mirror Equation Calculator, changing Mirror focal length, Object distance, and Object height rebuilds the numeric substitution for 1/(1/focalLength-1/objectDistance). The engine converts selected measurements to m, m, and m, retains unrounded values, and, when reverse solving is available, inserts the solved variable back into the same equation to verify 1/(1/focalLength-1/objectDistance) and Mirror Equation — image distance with a numerical residual.
Input and output interpretation
Use measured or documented values for Mirror focal length, Object distance, Object height. The calculated outputs are Mirror Equation — image distance, Mirror magnification, Mirror Equation — image height, Radius of curvature. Check each intermediate line before relying on the final value; an implausible intermediate quantity usually identifies a unit, range or assumption error.
Algorithm and verification
The Mirror Equation Calculator uses its own `mirror-equation` JavaScript engine to calculate 1/(1/focalLength-1/objectDistance) and Mirror Equation — image distance from Mirror focal length, Object distance, and Object height with 1/(1/focalLength-1/objectDistance). Calculator-owned boundary and mode tests check that workflow; an external frozen-reference oracle checks the numeric outputs, and physical source mutation testing confirms that a changed `mirror-equation` engine is rejected.
Visual interpretation
In the Mirror Equation Calculator, this guidance applies to Mirror focal length, Object distance, and Object height and the reported 1/(1/focalLength-1/objectDistance) and Mirror Equation — image distance. The `mirror-equation` workflow evaluates 1/(1/focalLength-1/objectDistance) in m, m, and m; Confirm sign conventions, unit systems, material properties and boundary conditions before engineering use.. Verification context: dbfb47.
Dimensional formula audit
The mirror equation calculator normalizes focalLength (Mirror focal length, m), objectDistance (Object distance, m), objectHeight (Object height, m) before evaluating the equations. Its reported quantities are Mirror Equation — image distance in m, Mirror magnification in x, Mirror Equation — image height in m, Radius of curvature in m. This separation matters because a numerical value without its measurement dimension can produce a plausible-looking but physically or financially incorrect answer. Conversion factors are applied before substitution, and output conversion occurs only after the canonical result has been calculated at full precision.
Mirror Equation — image distance = 1 ÷ (1 ÷ focalLength-1 ÷ objectDistance)Mirror magnification = -(1 ÷ (1 ÷ focalLength-1 ÷ objectDistance)) ÷ objectDistanceMirror Equation — image height = objectHeight × -(1 ÷ (1 ÷ focalLength-1 ÷ objectDistance)) ÷ objectDistanceRadius of curvature = 2 × focalLength
Formula sensitivity and boundary verification
To verify the mirror equation calculator, hold all other inputs fixed and change focalLength within its permitted range. The live substitution line shows exactly where that value enters the equation for Mirror Equation — image distance. Repeat the check with objectDistance. The result must follow the displayed algebra, remain finite, and retain the stated output unit. At minimum and maximum boundaries, the validator rejects undefined domains, impossible denominators and nonphysical values rather than silently returning a number.
Manual reproduction of the result
For an independent hand check, first convert every selected unit to the canonical units shown in the variable table. Next copy the governing equation, replace each symbol with the canonical value shown in the live derivation, and calculate the intermediate expression without early rounding. Finally round only once to the displayed precision and compare both the numerical value and unit with the calculator card. This process makes the mirror equation calculator reproducible instead of relying on an unexplained result.
Limitations
The Mirror Equation Calculator validates the supported domain for Mirror focal length, Object distance, and Object height before evaluating 1/(1/focalLength-1/objectDistance). Undefined denominators, impossible roots, non-finite values, and out-of-range selections are rejected before 1/(1/focalLength-1/objectDistance) and Mirror Equation — image distance is displayed.
What equations does the mirror equation calculator use?
Mirror Equation — image distance = 1 ÷ (1 ÷ focalLength-1 ÷ objectDistance); Mirror magnification = -(1 ÷ (1 ÷ focalLength-1 ÷ objectDistance)) ÷ objectDistance; Mirror Equation — image height = objectHeight × -(1 ÷ (1 ÷ focalLength-1 ÷ objectDistance)) ÷ objectDistance; Radius of curvature = 2 × focalLength
How are units handled?
For the Mirror Equation Calculator, Mirror focal length, Object distance, and Object height are normalized to m, m, and m before 1/(1/focalLength-1/objectDistance) is evaluated. The unrounded value used for 1/(1/focalLength-1/objectDistance) and Mirror Equation — image distance is retained internally; formatting is applied only to the result cards. This section’s cross-check centers on Object distance in the `mirror-equation` workflow.
How can the result be independently checked?
To reproduce the Mirror Equation Calculator independently, convert Mirror focal length, Object distance, and Object height to m, m, and m, substitute those canonical values into 1/(1/focalLength-1/objectDistance), keep full precision through the intermediate arithmetic, and round only the final 1/(1/focalLength-1/objectDistance) and Mirror Equation — image distance to the displayed precision.
Formula-based calculator guide
Mirror Equation Calculator: formula, steps, units and verification
mirror equation calculator is designed for a transparent calculation rather than a black-box answer. Mirror Equation Calculator: calculate image distance with the mirror equation calculator. Check measurement units, formula substitution, reverse solving.
How to use the mirror equation calculator in 5 steps
- Choose the required mode. Select the calculation path that matches the quantity you know and the result you need.
- Enter source values. Use measured, documented or assignment values rather than rounded estimates whenever possible.
- Confirm every unit. The mirror equation calculator converts supported units before formula substitution, so each selector must describe the entered number.
- Run the calculation. Review the displayed formula, normalized values and numeric substitution before accepting the final result.
- Verify the answer. Reproduce the substitution manually and check whether the output is reasonable for the stated assumptions.
Mirror Equation Calculator formula and unit checks
The mirror equation calculator keeps source inputs, canonical calculation units and displayed output units separate. This prevents a correct formula from producing a wrong answer because feet were treated as meters, percentages as decimals, or time values as the wrong interval.
For an independent check, copy the formula shown by the calculator, substitute the unrounded canonical values, preserve full precision through intermediate operations and round only the final result. The verified answer should match both the displayed number and its measurement unit.
How to interpret the mirror equation calculator result
A result is useful only when its assumptions match the real problem. Compare the answer with expected ranges, inspect any warning or boundary message, and test a nearby input to confirm that the output changes in the direction predicted by the governing relationship.
The mirror equation calculator provides an educational and planning result. For regulated, medical, structural, financial, laboratory or safety-critical decisions, verify the inputs and method against the applicable professional requirements before acting.
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Mirror Equation Calculator quick verification checklist
Before saving or reporting a result from the mirror equation calculator, confirm the input source, unit selections, formula mode, intermediate substitution, final unit and rounding rule. These checks make the calculation reproducible and easier to audit.
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