A curved mirror bends the direction of reflected light in a way that depends on where the light came from, and the mirror equation is the short arithmetic statement of that behaviour: the reciprocal of the focal length equals the reciprocal of the object distance plus the reciprocal of the image distance. Its companion, m = −dᵢ/d₋, converts those distances into how large the image is and which way up. Between them they answer every question a spherical mirror can raise.
This mirror equation calculator from Arb Digital takes any two of the three quantities and returns the third, along with the radius of curvature, the magnification and the image height. It also does the part that trips people up: it reads the signs back to you in words, saying whether the image is real or virtual, upright or inverted, enlarged or reduced, so that a negative number is interpreted rather than merely displayed.
What This Mirror Equation Calculator Does
You pick concave or convex, enter the curvature as a positive magnitude, and the tool applies the sign for you. This is deliberate. The most common failure in mirror problems is not the algebra but the convention, and asking for a signed focal length invites an error that produces a plausible-looking wrong answer. A concave mirror gets a positive focal length, a convex mirror a negative one, and the rest of the arithmetic follows.
Curvature can be supplied either as a focal length or as a radius of curvature, because sources quote both. For a spherical mirror they are related by R = 2f, so the tool converts whichever you give and reports both. Object and image distances then go in, and you leave exactly one of the three quantities blank for the calculator to find. Leave two blank and it says the problem is under-determined rather than guessing; fill all three and it says it cannot tell which one you meant to solve for.
Units are not fixed. The equation is homogeneous in length, so as long as every distance is in the same unit, the answer comes back in that unit. Centimetres are the default because that is how most optics problems are set, but metres or inches work identically. Magnification is a pure ratio and carries no unit at all.
How to Use It
- Choose the mirror type first. Concave mirrors curve away from the viewer at the edges and can form real images; convex mirrors bulge towards the viewer and never do.
- Enter the curvature as a positive number. Whether you type a focal length or a radius, give the magnitude. The tool signs it according to the mirror type you selected.
- Fill exactly two of the three distance-related boxes. Blank means unknown. Typing 0 is not the same thing: a zero object distance means the object is pressed against the mirror surface, which the tool will reject.
- Add an object height if you want a real size. It changes nothing in the optics; it only scales the magnification into a physical image height. Leave it out and the magnification alone is still correct.
- Read the note, not just the numbers. It converts the signs into plain statements about the image, which is what most questions are actually asking for.
The Formula and a Worked Example
The mirror equation is 1/f = 1/d₋ + 1/dᵢ, with f = R/2 and m = −dᵢ/d₋. The sign convention used here is the standard one: distances measured in front of the mirror, on the side the light is on, are positive; distances behind the mirror are negative. A concave mirror has a positive focal length, a convex mirror a negative one. A positive image distance describes a real image formed in front of the mirror; a negative one describes a virtual image apparently behind it.
Take the default: a concave mirror of 10 cm focal length with the object 30 cm away. Rearranged, 1/dᵢ = 1/10 − 1/30 = 1/15, so the image forms 15 cm in front of the mirror and is real. The magnification is −15/30 = −0.5, so the image is inverted and half the size, and a 5 cm object gives a 2.5 cm inverted image. The radius of curvature is 20 cm. The OpenStax University Physics section on spherical mirrors derives the equation from the geometry and uses the same convention.
The Five Cases a Concave Mirror Produces
A concave mirror behaves differently depending on where the object sits relative to the focal point and the centre of curvature, and knowing the five regions saves a lot of arithmetic. Beyond the centre of curvature, the image is real, inverted and reduced, and it lies between the focus and the centre. At the centre of curvature exactly, the image is real, inverted and the same size, formed at the same point — which is how you find R experimentally without any calculation at all.
Between the centre and the focus, the image is real, inverted and enlarged, and lies beyond the centre. This is the projector region. At the focus itself, the reflected rays leave parallel and no image forms at any finite distance; the calculator reports this as a genuine limit rather than dividing by zero. Inside the focus, the image becomes virtual, upright and enlarged — the shaving-mirror or make-up-mirror case, and the only concave configuration where you see yourself the right way up and larger.
A convex mirror has no such variety. Its focal length is negative, so 1/dᵢ is always negative for any real object, which means the image is always virtual, always upright and always reduced. That uniformity is why convex mirrors are used for wide-angle security and vehicle applications: the field of view is wide and the behaviour is predictable at every distance. It is also why the warning about objects being closer than they appear exists, since the reduction fools the brain's distance estimate.
Mirrors and Lenses Use the Same Equation and Different Geography
The thin lens equation is identical in form to the mirror equation, which is convenient and also a trap. The difference is where the image forms relative to the light. A real image from a mirror appears on the same side as the object, because reflected light goes back the way it came. A real image from a lens appears on the far side, because the light continues through. Any sign convention that handles both has to encode that difference somewhere, and the encoding is what varies between textbooks.
There is also a difference in what curvature means. A mirror's focal length depends only on its shape, through f = R/2. A lens's focal length depends on both its shape and the refractive index of its material, which is what the lens maker's equation adds. That is why a lens changes focal length when you put it underwater and a mirror does not, and it is the reason underwater photographers use flat ports rather than assuming their lens will behave. The thin lens equation calculator and the lens maker equation calculator handle those two steps, and the OpenStax section on thin lenses sets both out.
Where the Equation Stops Being Accurate
Everything on this page rests on the paraxial approximation: rays are assumed to stay close to the optical axis and to make small angles with it, so that the sine of an angle can be replaced by the angle itself. Real mirrors are used with rays that are not always so well behaved, and the errors have names.
Spherical aberration is the main one. A spherical mirror does not bring off-axis parallel rays to exactly the same point as near-axis rays, so a wide mirror produces a blurred rather than a point focus. The severity grows quickly with aperture, which is why serious reflecting telescopes use parabolic rather than spherical primaries; a paraboloid focuses all parallel axial rays exactly, and the mirror equation ceases to apply in its simple form. Astigmatism and coma appear for objects off the axis, and neither is captured by a one-dimensional equation.
The equation also assumes a single reflecting surface in air and an object that is a point or a flat plane perpendicular to the axis. Compound systems must be worked through one surface at a time, with the image from each surface becoming the object for the next, and a virtual image feeding forward as a negative object distance. This calculator handles one surface; chain it manually for more.
How This Differs From the Adjacent Arb Digital Tools
This page is specifically for reflecting surfaces, and it takes a radius of curvature as an alternative input because that is how mirrors are usually specified. The thin lens equation calculator solves the same algebraic relation for refracting elements, where the sign convention for image location differs. The diopter calculator works in reciprocal metres rather than focal lengths, which is the unit prescriptions use. The Snell's law calculator and the index of refraction calculator deal with the boundary between two media, which a mirror does not have. And the Malus law calculator covers what happens to intensity through a polariser, a property this page ignores completely since ray optics treats light as directionless energy.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Entering a negative focal length for a convex mirror — select the mirror type and give the magnitude. Signing it twice turns a convex mirror back into a concave one.
- Confusing the focal length with the radius of curvature — R is exactly twice f for a spherical mirror, so mixing them up doubles or halves every distance in the answer.
- Reading a negative magnification as an error — the minus sign means inverted, not wrong. Magnitude gives the size ratio, sign gives the orientation.
- Typing 0 for a distance you do not know — blank means unknown. A zero object distance places the object on the mirror surface, which has no meaningful solution.
- Applying the equation to a wide-aperture mirror — the paraxial approximation degrades quickly as the mirror gets large relative to its focal length, and spherical aberration then dominates.
Related Free Tools From Arb Digital
Move from mirrors to lenses with the thin lens equation calculator and the lens maker equation calculator, or into prescription units with the diopter calculator. For refraction at a boundary use the Snell's law calculator and the index of refraction calculator. Polarisation is handled by the Malus law calculator, and angle units by the angle converter. The full free online tools hub lists everything Arb Digital has published.
Frequently Asked Questions
One over the focal length equals one over the object distance plus one over the image distance. Together with the magnification relation, in which magnification is minus the image distance divided by the object distance, it describes every image a spherical mirror can form.
Because its focus lies behind the reflecting surface rather than in front of it, and the convention makes distances behind the mirror negative. The consequence is that a convex mirror can only ever produce virtual, upright, reduced images.
That the image forms behind the mirror and is virtual. You can see it by looking into the mirror, but you cannot catch it on a card or a sensor placed where it appears to be, because no light actually converges there.
For a spherical mirror the radius is exactly twice the focal length. This follows from the geometry under the paraxial approximation, and it is the reason an object placed at the centre of curvature forms a same-size image at the same point.
The reflected rays leave the mirror parallel to one another and never converge, so no image forms at any finite distance. The arithmetic shows this as a division by zero, and the calculator reports the limiting behaviour rather than an error code.
Because your face is inside the focal length of a concave mirror. In that region alone a concave mirror produces a virtual, upright, magnified image. Move further back than the focus and the image flips and becomes real.
Only approximately. The equation assumes a spherical surface used near its axis. A paraboloid focuses parallel axial rays perfectly and behaves differently for other rays, so telescope optics need a fuller treatment than a single reciprocal relation.
This tool is provided for educational and study use. It applies the paraxial spherical-mirror relations as written and is not a substitute for full optical design software when aberrations matter.
