Concave and convex mirror surfaces curve in opposite directions: a concave mirror curves inward and converges reflected light to a real focus, while a convex mirror bulges outward and makes light diverge from a virtual focus behind it. Both obey the mirror equation: one over f equals one over v plus one over u. Convex images are always virtual.
Pick up a metal spoon and look at your face in the hollow side. You are upside down. Turn the spoon over, look at the back, and there you are again — the right way up, small, and slightly squashed. Same spoon. Same light. Same eyes.
Nothing about you changed. What changed was which way the reflecting surface curved — and that single geometric fact decides whether light comes together or spreads apart. It is the whole of this article in one piece of cutlery.
What Is a Concave and Convex Mirror?
Start with the shape, because everything else follows from it. Both mirrors are cut from the surface of an imaginary sphere; the only question is which side of that sphere is silvered.
A concave mirror is silvered on the outside of the sphere, so the reflecting surface caves inward, away from you. Rays arriving parallel to the axis are folded toward one another and cross at a genuine point in space.
A convex mirror is silvered on the inside, so the reflecting surface bulges outward, toward you. Parallel rays are thrown apart. Trace them backwards and they seem to stream out of a point behind the glass — a point no light ever actually reaches.
That is the entire distinction. A concave mirror is a converging mirror; a convex mirror is a diverging mirror.
Parallel light on a concave and convex mirror. Solid gold lines are real rays; dashed lines are backward extensions that never carry energy.
A mnemonic that actually survives the exam
Concave caves in. Stand a torch in front of one and the beam is pulled together; stand in front of a convex one and the beam is pushed away.
If you would rather anchor it to something you own: the inside bowl of a spoon is concave, the polished back is convex. Everything below is that spoon, done with algebra.
Concave and Convex Mirror: 6 Key Differences at a Glance
Six properties separate the two mirrors, and the first one drives all the others.
| Property | Concave mirror | Convex mirror |
|---|---|---|
| 1. Surface shape | Curves inward (caves away from the object) | Bulges outward (toward the object) |
| 2. Effect on parallel rays | Converges them | Diverges them |
| 3. Focus and sign of f | Real focus in front; f is positive | Virtual focus behind; f is negative |
| 4. Image type possible | Real or virtual, depending on where the object sits | Virtual only — always, with no exception |
| 5. Orientation and size | Inverted and diminished, same-size, or magnified — six distinct cases | Upright and diminished, every single time |
| 6. Field of view | Narrow | Wide — the reason it is a safety mirror |
| Best uses | Shaving mirrors, dentist’s mirrors, headlamp reflectors, solar cookers, telescope primaries | Passenger-side wing mirrors, shop security mirrors, blind-corner road mirrors, ATM mirrors |
Row 4 is the one worth memorising. A convex mirror producing a real image is not a hard problem — it is an impossible one, and later on we will prove it in three lines of algebra.
The Mirror Formula: 1/f = 1/v + 1/u
One equation covers both mirrors. Nothing switches; only the sign of f changes.
The focal length is fixed by the sphere the mirror was cut from:
And the magnification links the two heights to the two distances:
Every symbol, with its unit
| Symbol | Quantity | SI unit | Sign rule |
|---|---|---|---|
| u | Object distance from the pole | metre (m) | Always positive for a real object |
| v | Image distance from the pole | metre (m) | Positive if in front (real); negative if behind (virtual) |
| f | Focal length | metre (m) | Positive for concave; negative for convex |
| R | Radius of curvature | metre (m) | Same sign as f, since R = 2f |
| ho | Object height | metre (m) | Taken as positive (upright object) |
| hi | Image height | metre (m) | Positive if upright; negative if inverted |
| m | Linear magnification | dimensionless | m > 0 upright; m < 0 inverted; |m| > 1 magnified |
Any consistent unit works — centimetres throughout, or metres throughout. Mixing them is the single most common arithmetic disaster in this topic.
Pin the sign convention down before you substitute
Different textbooks measure distances from different origins, and both give correct answers if you never mix them. This article uses the real-is-positive convention above, which is also what our calculator uses.
In practice, the slip that costs marks is subtle. A student half-remembers a Cartesian convention where u is negative, writes u = −30 cm, but then also writes f = +10 cm for a concave mirror out of habit. Two conventions in one equation, and the two errors do not cancel.
Choose one convention. Write the signs down before you touch the algebra. You can also check any answer instantly with our Lens & Mirror Calculator, which solves the mirror equation for focal length, either distance or the magnification and shows every step of the working.
Why is the focal length exactly half the radius?
Take a ray parallel to the axis, striking the mirror at a height h above it. The normal at that point runs along the radius, straight back to the centre of curvature C.
The law of reflection makes the reflected ray leave at the same angle to that normal. Simple geometry then puts the crossing point on the axis at exactly half of R — but only when h is small compared with R, so that the angles stay small.
This is the paraxial approximation, and f = R/2 lives or dies by it. Rays that strike far from the axis cross the axis a little closer to the mirror, smearing the focus. Opticians call that spherical aberration, and it is precisely why serious telescope mirrors are ground into paraboloids rather than spheres.
How Concave and Convex Mirrors Form Images
You do not need a protractor. Three standard rays are enough, and any two of them fix the image.
- Ray 1 — parallel in, through F out. A ray arriving parallel to the axis reflects through the focus (concave) or appears to come from the focus (convex).
- Ray 2 — through F in, parallel out. The reverse of ray 1, because light paths are reversible.
- Ray 3 — strike the pole. At the pole the mirror is locally flat and the axis is the normal, so the ray reflects at an equal angle on the other side of the axis.
Where the reflected rays actually meet, you get a real image — catchable on a screen. Where only their backward extensions meet, you get a virtual image — visible to the eye, but with no light at that location.
Ray diagrams for a concave and convex mirror. Solid rays carry light; dashed lines are extensions behind the convex mirror, where the virtual image sits.
The six cases of a concave mirror
Slide the object along the axis and the image changes character six times. This table is worth knowing cold.
| Object position | Image position | Nature | Orientation | Size |
|---|---|---|---|---|
| At infinity | At F | Real | Inverted | A point |
| Beyond C (u > 2f) | Between F and C | Real | Inverted | Diminished |
| At C (u = 2f) | At C | Real | Inverted | Same size |
| Between C and F | Beyond C | Real | Inverted | Magnified |
| At F (u = f) | At infinity | No image forms | — | — |
| Between F and P (u < f) | Behind the mirror | Virtual | Upright | Magnified |
A convex mirror needs no such table. Its image is always virtual, always upright, always smaller, and always squeezed into the gap between the pole and the focus.
Why a Convex Mirror Can Never Form a Real Image
Most articles simply assert this. It is more satisfying to watch it fall out of the equation, and it takes three lines.
Write the focal length of a convex mirror as f = −F, where F is a positive number. Rearrange the mirror formula for v:
Invert it:
Now read what that says. For any real object, u is positive and F is positive, so v is negative no matter what you choose. Negative v means the image sits behind the mirror. Virtual. Always.
Push it one step further and substitute into m = −v/u:
The denominator is always bigger than the numerator, so m is always positive and always less than 1. Positive means upright; less than one means diminished. There is no object distance, no mirror, no trick that escapes this.
Notice too that |v| = uF/(u + F) is always smaller than F. The image is trapped between the pole and the virtual focus — which is why a convex mirror can pack an entire car park into a palm-sized reflection.
Real-World Examples of Concave and Convex Mirrors
1. Your passenger-side wing mirror
That mirror is convex, and its curvature is not left to the designer’s taste. Under the US federal standard FMVSS 111, a convex rearview mirror must have an average radius of curvature between 889 mm and 1,651 mm, and must be permanently marked with the warning that objects are closer than they appear.
Run those numbers through f = R/2 and the focal length lands between about −44 cm and −83 cm. Worked Problem 4 below uses a mirror right in that band, and the image of a following car comes out at roughly 7% of full size.
Your brain has one rule for judging distance from a familiar object: small means far. The mirror shrinks the car, your brain reports “far”, and the etched warning is there to overrule you.
2. The shaving and make-up mirror
Concave, with a focal length of roughly 20–40 cm. Bring your face closer than the focal point and you land in the sixth row of the table: virtual, upright, magnified.
Step slowly backwards past the focus and your reflection blurs, flips, and reappears upside down. That is the moment u crosses f, and it is the cheapest optics experiment in the house.
3. The dentist’s mirror
Same physics, smaller focal length. The mirror is held a centimetre or two from a tooth — well inside the focus — so the dentist gets an upright, magnified virtual image without contorting their neck.
4. Headlamps, torches and solar cookers
These run the concave mirror backwards. Put the light source at the focus and every ray leaves parallel to the axis: a beam instead of a glow.
Reverse it again and you have a solar cooker, gathering parallel sunlight onto a pot sitting at the focus. Reversibility of light paths is doing all the work here.
5. Telescopes — where both mirrors appear at once
The James Webb Space Telescope is the cleanest example of the two mirrors cooperating. Its 6.5-metre segmented primary is concave, and it focuses light onto a smaller convex secondary mirror at the end of the booms, which sends it back through a hole toward the instruments.
The concave mirror gathers and converges; the convex one spreads the converging cone slightly, lengthening the effective focal length without lengthening the telescope. A whole observatory, folded up by curvature.
Common Misconceptions About Concave and Convex Mirrors
Misconception 1: “A concave mirror always magnifies”
It magnifies only when the object is closer than C. Put the object beyond the centre of curvature and the image is real, inverted and smaller — that is exactly how a telescope primary works on a distant star.
The magnifying behaviour people remember is one row of a six-row table, not the whole story.
Misconception 2: “The focal length equals the radius of curvature”
It is half of it: f = R/2. A concave mirror ground from a sphere of radius 36 cm has a focal length of 18 cm, not 36 cm.
Students who forget the factor of two usually get an image distance that is wrong by a factor of roughly two as well — a satisfyingly detectable error, if you sanity-check.
Misconception 3: “A virtual image isn’t really there, so a camera can’t photograph it”
Point your phone at a convex security mirror and it captures the image perfectly. “Virtual” does not mean imaginary; it means no light physically converges at the image location.
The rays leaving the mirror genuinely diverge from that point, so any lens — your eye’s, your camera’s — can focus them into a real image on a retina or a sensor. What you cannot do is hold a paper screen at that spot and catch a picture.
Misconception 4: “Convex mirrors shrink things because the mirror is small”
Size has nothing to do with it. Build a convex mirror three metres across and the image is still diminished, because m = F/(u + F) is less than 1 for every possible u.
A large plane mirror shows objects at full size. It is the curvature, not the area, that does the shrinking — and the shrinking is what buys the wide field of view.
How Mirrors Relate to Lenses, Light and Reflection
The mirror equation and the thin-lens equation are the same equation. Both read 1/f = 1/v + 1/u; a converging lens behaves like a concave mirror, a diverging lens like a convex one. Learn one sign convention properly and you have learned two topics.
Underneath the geometry sits a wave. Light is a transverse wave, and reflection leaves its frequency untouched — which is why your reflection is not a different colour from you.
Because the wave travels at the speed of light, the image appears instantaneously for all practical purposes. Over the 6.5 metres of Webb’s primary mirror, the path difference costs about 20 nanoseconds.
Move the mirror, though, and the frequency does shift. That is the Doppler effect operating on reflected light, and it is exactly how a police radar gun measures the speed of the car whose convex wing mirror we started with.
