Mirror Equation Calculator
Solve spherical and plane mirror relationships with the Cartesian sign convention, including image distance, focal length, radius of curvature, and magnification.
Known quantities
For a curved mirror, edit any two compatible numeric fields. The two most recently edited quantities become the known values; the rest are recalculated. A plane mirror needs only object distance.
Live results
How to use this mirror equation calculator
What this calculator does
This calculator solves the core geometric relationships for concave, convex, and plane mirrors. For curved mirrors it combines the mirror equation 1/f = 1/v + 1/u, the spherical-mirror relationship r = 2f, linear magnification m = – v/u, and areal magnification m². It is designed to solve one consistent paraxial mirror model, not to trace off-axis rays, model spherical aberration, or determine whether a real optical system meets manufacturing tolerances. The calculator uses the same Cartesian sign pattern shown beside the fields: an ordinary object in front of the mirror has a negative object distance; concave focal length and radius are negative; convex focal length and radius are positive; and image distance can be positive or negative.
When to use it
Use it to check a classroom mirror-equation problem, convert between focal length and radius of curvature, infer image position from an object position and mirror focal length, or determine magnification when you know two compatible quantities. It is also useful for comparing the sign behavior of concave and convex mirrors. For a conceptual review of image formation and the paraxial mirror equation, see the OpenStax section on spherical mirrors.
How to calculate
- The calculator opens with a complete demonstration: a concave mirror, Object distance ( – ) = – 6 cm, and Focal length ( – ) = – 12 cm. The matching workbook is immediately available through Download Excel.
- Choose Mirror type. For a curved mirror, edit any two compatible numeric fields. The two most recently edited fields become the known quantities and are marked “Known”; all other numeric fields are recalculated from the same canonical model. If you edit a third field, it replaces the older known quantity.
- Read the values in Live results. The large result emphasizes Image distance, while the remaining cards repeat Object distance, Focal length, Radius of curvature, Linear magnification, and Areal magnification from the same calculation.
- Select Download Excel to create a current-state .xlsx workbook. Reset is deliberately different from the startup example: it clears the numeric data, restores Concave mirror as the neutral mirror type, clears calculated results, and disables export until a complete valid state is entered again.
Input guide
Mirror type is required and accepts Concave mirror, Convex mirror, or Plane mirror. Concave and convex mirrors require two independent finite numeric quantities; plane mirrors require Object distance only because their image distance and magnifications are fixed by symmetry. Changing the mirror type also changes the expected sign of Focal length and Radius of curvature.
Object distance ( – ) is a required distance in centimeters whenever it is one of the known quantities. Enter a finite negative decimal such as – 6. Commas are accepted only as conventional thousands separators, so 1,000 is valid while decimal-comma text such as 1,5 is rejected rather than reinterpreted. A common mistake is entering the positive physical distance instead of its Cartesian signed value.
Image distance (+/ – ) is a finite distance in centimeters and may be used as one of the two known quantities. In this calculator's convention, a positive value lies behind the mirror and corresponds to a virtual image; a negative value lies in front of the mirror and corresponds to a real image. A value of zero is excluded because it makes the reciprocal mirror equation undefined.
Focal length ( – ) in the startup concave mode is measured in centimeters and must be negative; when Mirror type is Convex mirror, the label changes to Focal length (+) and the value must be positive. A realistic example is – 12 cm for the startup concave mirror. A focal length cannot be zero. If Object distance equals Focal length for a curved mirror, the ideal paraxial model places the image at a non-finite limit, so this calculator asks for a nearby finite case instead of pushing a non-finite number into the page or workbook.
Radius of curvature ( – ) is another centimeter input for the startup concave mode and becomes Radius of curvature (+) for a convex mirror. It is exactly twice the focal length for the spherical-mirror model, so Focal length and Radius of curvature are not an independent pair by themselves. For the startup case, r = – 24 cm. A common mistake is trying to use focal length and radius together as the only two known quantities; the calculator rejects that underdetermined pair.
Areal magnification (+) is a required positive, dimensionless ratio only when you choose it as a known quantity. A value of 4 means the image area is four times the object area in the idealized model. Because areal magnification is the square of linear magnification, it contains no orientation sign. Some combinations can therefore be ambiguous; if both upright and inverted solutions satisfy the same two known values, the calculator asks you to choose a signed or distance-based quantity instead.
Linear magnification (+/ – ) is a signed, dimensionless ratio. The startup value is 2. Positive magnification indicates an upright image in this convention; negative magnification indicates an inverted image. A magnitude greater than one means enlargement and a magnitude below one means reduction. Zero is not accepted as a finite mirror-equation solution.
Output guide
Image distance is the highlighted result in centimeters and is driven by whichever compatible pair is currently known. Object distance, Focal length, and Radius of curvature are the corresponding signed distances from the same model. Linear magnification is the signed height ratio, while Areal magnification is its nonnegative square and therefore describes area scale without orientation. These are mathematical estimates from the paraxial mirror model, not measurements or manufacturing recommendations. A high absolute magnification means a larger modeled image; values near one imply similar linear size; and a sign change in linear magnification reflects a change in orientation.
Worked example
The first-open demonstration uses a concave mirror with u = – 6 cm and f = – 12 cm. Rearranging the mirror equation gives 1/v = 1/f – 1/u = – 1/12 – ( – 1/6) = 1/12, so v = 12 cm. The radius is r = 2f = – 24 cm. Linear magnification is m = – v/u = 2, and areal magnification is m² = 4. Those exact values appear in the startup controls, result cards, and workbook checkpoints.
Learn more
The compact formula list in OpenStax University Physics key equations gives the mirror equation, spherical-mirror focal relation, and magnification identity in one place. Sign conventions can differ between textbooks, so keep one convention consistent from inputs through interpretation rather than mixing signs from different sources.
Understanding the mirror model
The mirror equation is most accurate for rays close to the principal axis, where the paraxial approximation makes a spherical mirror behave like an ideal converging or diverging element. The radius relation f = r/2 links the geometry of a spherical surface to its focal length. The Boston University notes on spherical mirrors provide worked examples connecting image distance, focal length, and magnification.
For a plane mirror, the ideal focal length and radius do not have finite values. Instead of placing a non-finite token into numeric outputs or the exported workbook, this calculator reports those two quantities as “No finite value” while still returning the finite identities v = – u, linear magnification 1, and areal magnification 1.