Mirror Formula Calculator: Concave and Convex Mirror Ray Diagrams

Pick up a spoon and look at the inside of the bowl at arm's length. You are upside down. Bring it slowly toward your face and, somewhere along the way, you flip the right way up and swell to fill the whole bowl. Turn the spoon over and look at the back: you are upright, tiny, and stay that way no matter how you move it. Nothing about the spoon changed between those three views — only the distance, and which way the metal curves.

One equation covers all of it. The mirror formula, $1/v + 1/u = 1/f$, takes the object distance and the mirror's focal length and returns exactly where the image sits, which way up it is, and how big it is. This page walks through the sign convention that makes it work, the three construction rays that draw it, and the five zones of a concave mirror that every exam question is secretly about — with three simulations to drag around: one showing what a focus physically is, one live ray-diagram bench you drag the object across, and one overhead view of why every car mirror bulges.

What Is the Mirror Formula and What Do u, v and f Mean?

$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Three distances, all measured from the pole — the point where the mirror's surface crosses its own axis. The formula is useless without a sign convention, and the one used in NCERT, JEE and NEET is the New Cartesian convention: light travels toward the mirror in the positive direction, and any distance measured against that direction is negative.

The single line worth memorising above all the others is the one about $v$. For a mirror, a real image has $v < 0$. Real light comes back out of a mirror on the same side it went in, so anything real forms in front of the glass. That is the exact opposite of a thin lens, where a real image sits on the far side and $v$ is positive. Students who learn lenses first and then assume mirrors behave the same way get every real/virtual answer backwards.

The relation $R = 2f$ says the focus sits exactly halfway between the pole and the centre of curvature. It is not a definition — it is a result, and a slightly shaky one, as the next section shows.

How Do You Calculate the Image Position Step by Step?

Take a concave mirror of focal length 15 cm with an object 40 cm in front of it — the setting the simulator further down starts on.

1. Attach the signs. Concave, so $f = -15$ cm. Object in front, so $u = -40$ cm.
2. Rearrange for v. $\dfrac{1}{v} = \dfrac{1}{f} - \dfrac{1}{u} = \dfrac{1}{-15} - \dfrac{1}{-40}$
3. Do the arithmetic. $\dfrac{1}{v} = -\dfrac{8}{120} + \dfrac{3}{120} = -\dfrac{5}{120} = -\dfrac{1}{24}$, so $v = -24$ cm.
4. Read the sign. $v$ is negative, so the image is real, 24 cm in front of the mirror.
5. Magnify. $m = -\dfrac{v}{u} = -\dfrac{-24}{-40} = -0.60$ — negative, so inverted, and $|m| < 1$, so diminished.

Real, inverted, and 60% of the object's height. Every mirror question is these five steps.

What Does the Focus of a Concave Mirror Actually Do?

Most diagrams reduce the focus to a letter F on a line, which makes it look like a label rather than a place. It is a place, and the way to see it is to send in a wall of light travelling parallel to the axis — light from something effectively infinitely far away, like the sun or a star — and watch where the reflected rays end up.

The simulation below does exactly that, ray by ray, off a genuine curved surface. Nothing in it uses the mirror formula: each ray is reflected about the true surface normal and followed until it meets the axis. The dish has a fixed radius of curvature of R = 100 cm, so the textbook answer for its focus is R/2 = 50 cm. The aperture slider controls how far off-axis the outermost ray is allowed to strike.

So $f = R/2$ is a paraxial result. It is the limit as the rays crowd in toward the axis, and it degrades smoothly as you open the dish up — 4.83% out at an aperture of 0.30 × R, nearly 12% out at 0.45 × R. The blur has a name, spherical aberration, and the curve the light piles up along is the caustic — the same bright cusp you can see in a mug of coffee under a desk lamp. Grinding the surface to a parabola instead of a sphere removes it completely, which is why a Newtonian telescope's primary is figured to a parabola rather than left spherical. The general cure is always a non-spherical figure, not the parabola specifically: most large modern telescopes are Ritchey–Chrétien designs whose primary is a hyperboloid, chosen because it cancels coma as well. A cheap torch reflector is figured to nothing at all, and keeps the aberration.

The Direction toggle is worth a second look, because it is the same dish doing two jobs. Light paths run equally well backwards: a mirror that gathers parallel light into a point will, fed from that point, throw a parallel beam back out. Collect is a telescope or a solar cooker. Project is a headlight or a searchlight. One piece of geometry, arrows reversed — and on a sphere, imperfect in both directions for exactly the same reason.

How Are Ray Diagrams Constructed for a Curved Mirror?

You never need to trace hundreds of rays. Any two rays whose behaviour you already know will cross at the image point, and a curved mirror hands you three such rays for free.

Rule 1 — parallel in, through F out. A ray arriving parallel to the axis is reflected through the focus. This is the definition of the focus, run forwards.

Rule 2 — through F in, parallel out. A ray heading for the focus comes back parallel to the axis. This is Rule 1 run backwards, and it is the projector case from the simulation above.

Rule 3 — through C in, straight back out. A ray aimed at the centre of curvature hits the surface along the normal and retraces its own path exactly.

Why does the ray through C retrace itself?

Because every normal to a sphere is a radius. A ray aimed at C is travelling straight down a radius, so it meets the surface head-on at 0° incidence, and a ray at 0° incidence reflects straight back the way it came. That is the one construction rule with no lens counterpart at all — a lens has two surfaces with two separate centres of curvature, and a ray aimed at either one is still bent onward by the other, so no aimed ray ever retraces itself — and it is the reason the object, C, and the image always lie on one straight line.

Two of these three rays already fix the image. The third is a free check: if it does not pass through the same point, you have made an arithmetic error somewhere.

Interactive Concave and Convex Mirror Ray Diagram Simulator

Here all three rules run at once on a live bench. Drag the blue object arrow along the axis with your pointer and watch the construction rebuild itself, with light pulses travelling out to the mirror and coming back — the one thing that distinguishes a mirror from a lens. The |u| slider does the same job for keyboard users. The readout underneath is a live mirror-formula calculator: signed u, v, f, R and m, plus the verdict on the image.

Try switching individual rays off with the three chips. Any two of them already pin the image down, and following one at a time is far easier than untangling three. The screen test is the part worth dwelling on: a real image is made of light that genuinely converges at that spot, so a piece of card put there catches a picture. A virtual image has no such place — the light diverges after reflection and only appears to come from behind the glass. The card stays blank wherever you put it. That physical difference is what the sign of $v$ is recording, and it is why "real or virtual" is not bookkeeping.

What Is Magnification in a Spherical Mirror?

$m = -\frac{v}{u} = \frac{h'}{h}$

The minus sign is part of the mirror convention, not a typo — it is what makes a real image (both $u$ and $v$ negative) come out with a negative $m$, and therefore inverted. Reading a magnification is two independent questions:

• The sign tells you which way up. $m < 0$ means inverted, $m > 0$ means upright. Every real mirror image is inverted; every virtual one is upright. There are no exceptions in either direction.
• The size tells you how big. $|m| > 1$ is magnified, $|m| < 1$ is diminished, $|m| = 1$ is life-size.

So $m = -0.60$ is inverted and 60% as tall; $m = +3$ is upright and three times as tall; $m = +1/6$ is upright and a sixth of the size. Because $m = h'/h$ as well, magnification is also the fastest route to an image height once you have $v$: multiply the object height by $m$ and keep the sign.

How Is the Mirror Formula Different from the Lens Formula?

The two equations look almost identical, which is exactly the trap. Compare them against the lens formula and its interactive ray diagram:

The deeper difference is the mechanism. A mirror obeys the law of reflection: angle of incidence equals angle of reflection, at every point, exactly, with no dependence on wavelength. A lens bends light by refraction, which is governed by Snell's law and the refractive indices of the two media — and because refractive index varies with colour, a simple lens smears white light into a fringe of colour. A mirror never does. That is one of three reasons every research telescope built in the last century collects light with a mirror rather than a lens: a large lens also has to be supported at its rim, where it sags under its own weight, and it needs two flawless optical surfaces cut into a large, perfectly homogeneous blank of glass. A mirror is supported across its whole back, and only one surface has to be right. The last great refractor, the Yerkes 40-inch, was finished in 1897.

Why Do Concave and Convex Mirrors Form Different Kinds of Images?

A concave mirror converges light, so whether the reflected rays actually manage to cross depends on how much convergence they were given — which depends entirely on where the object sits. A convex mirror diverges light, so they never cross at all. That is the whole story, and it produces five distinct outcomes for a concave mirror and exactly one for a convex mirror.

The five zones of a concave mirror

Worked here with $f = -10$ cm, so $R = -20$ cm — the focus 10 cm from the pole and the centre of curvature 20 cm from it.

Read down the v column and the pattern is one continuous motion, not five separate cases. As the object walks in from far away, the image walks out — they cross at C, where both sit 20 cm from the mirror at the same size. Keep going and the image races off toward infinity, arriving there exactly as the object reaches F. Push past F and it reappears on the other side of the mirror, upright and enlarged. Every one of these rows is a setting you can dial up on the bench above.

Why is a convex mirror's image always virtual and diminished?

This one is a theorem, not a survey of examples. For a convex mirror $f > 0$, and the object is always in front, so $u < 0$. Then

$\frac{1}{v} = \frac{1}{f} - \frac{1}{u}$

is the sum of two strictly positive terms: written with magnitudes, $1/v = 1/f + 1/|u|$. Two consequences follow immediately and admit no exceptions.

• $1/v > 1/f$, so $0 < v < f$. Positive $v$ puts the image behind the mirror, so it is always virtual — and always upright, since $m = -v/u$ is then positive.
• $1/v > 1/|u|$, so $v < |u|$, so $|m| = v/|u| < 1$. It is always diminished.

There is no zone structure to learn, because there are no zones. Note also that $v < f$ always: no matter how far away the object, the image never gets further from a convex mirror than its own focal length.

Where Are Concave and Convex Mirrors Used in Real Life?

Concave mirrors show up wherever light needs gathering or throwing: telescope primaries, solar cookers, headlight and torch reflectors, dentists' mirrors, make-up and shaving mirrors. Convex mirrors show up wherever coverage matters more than size: vehicle door mirrors, blind-corner road mirrors, shop security domes, ATM surrounds.

The convex case deserves a proper look, because "it shrinks things" sounds like a defect rather than the point. The simulation below is an overhead plan view of a driver, a mirror, and a road — not a ray diagram. The driver's eye is fixed 60 cm in front of the glass, on the road side; the shaded wedge is the region the mirror actually shows them, and three markers on the road light up green when they fall inside it. The wedge is a paraxial construction, so the very widest settings are indicative rather than exact.

The mechanism is the one thing the plan view makes obvious and a ray diagram never does. Bulging the glass forms a virtual image of the driver's own eye closer to the mirror — 30 cm behind it instead of the flat mirror's 60 cm — and the cone drawn from that point out through the two rims opens up accordingly. The widened view and the shrunken image are not two effects to be traded off against each other. They are the same shrunken eye-image, seen twice. That is precisely what the warning printed on the glass means: objects in mirror are closer than they appear.

Worked Examples for Physics Exams

Mirror Formula and Sign Convention Quick Reference