Derivation of Mirror Formula: 1/v + 1/u = 1/f With Examples

The derivation of mirror formula ends in $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ , which links the object distance u, the image distance v and the focal length f of a spherical mirror. It comes from two pairs of similar triangles in a ray diagram and works for concave and convex mirrors alike, as long as you follow a sign convention.

Signs here follow the New Cartesian convention taught in the Indian syllabus for both the light chapter of Class 10 and ray optics in Class 12. All distances are measured from the pole P. Distances measured in the direction of the incoming light are positive, and those against it are negative. Light travels from the object toward the mirror, so the object lies against the direction of incident light and u is negative. Concave mirrors therefore carry a negative focal length and convex mirrors a positive one. The focal length is half the radius of curvature, $f = \frac{R}{2}$ .

Quick facts

Mirror formula
$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$
Focal length
$f = \frac{R}{2}$
Concave mirror
f negative, real images have v negative
Convex mirror
f positive, image always virtual with v positive
Unit
Any length unit, used the same for u, v and f (SI: metre)
US form
$\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$ with real distances positive
Magnification
$m = -\frac{v}{u}$

Formula of mirror and its symbols

Terms in the spherical mirror equation
Symbol	Meaning	Sign for a concave mirror
$u$	Object distance from the pole	Negative
$v$	Image distance from the pole	Negative if real, positive if virtual
$f$	Focal length, pole to principal focus F	Negative
$R$	Radius of curvature, pole to centre C	Negative

For a convex mirror f and R are positive, and the image always forms behind the mirror, so v is positive. The formula itself never changes. Only the signs of the numbers you put into it change.

Derivation of mirror formula, class 12 method

Take a concave mirror with pole P, focus F and centre of curvature C. An object AB stands on the principal axis beyond C, with A on the axis and B at the top. Its real, inverted image A′B′ forms between C and F. Draw two rays from B:

a ray parallel to the axis, which strikes the mirror at M and reflects through F;
a ray to the pole P, which reflects at an equal angle to the axis.

Drop a perpendicular from M to the axis. For a small mirror aperture its foot is very close to P, so treat PM as a straight line through P of length AB.

Step 1. Triangles A′B′F and PMF are similar (the angles at F are vertically opposite, and both triangles have a right angle):

$\frac{A'B'}{PM} = \frac{FA'}{FP} \quad\Rightarrow\quad \frac{A'B'}{AB} = \frac{FA'}{FP}$

Step 2. Triangles A′B′P and ABP are similar, because the angles of incidence and reflection at P are equal:

$\frac{A'B'}{AB} = \frac{PA'}{PA}$

Step 3. Equate the two right hand sides and write FA′ = PA′ − PF:

$\frac{PA' - PF}{PF} = \frac{PA'}{PA}$

Step 4. Apply the sign convention. All three points lie in front of the mirror, so PA′ = −v, PF = −f and PA = −u:

$\frac{-v + f}{-f} = \frac{-v}{-u} \quad\Rightarrow\quad \frac{v - f}{f} = \frac{v}{u}$

Step 5. Cross multiply to get $uv - uf = vf$ , then divide every term by $uvf$ :

$\frac{1}{f} - \frac{1}{v} = \frac{1}{u} \quad\Rightarrow\quad \frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Boards give marks for the labelled ray diagram, the two similar triangle statements and the sign substitution. Do not skip Step 4.

Worked numericals of mirror formula

Example 1: concave mirror, real image

An object is placed 25 cm in front of a concave mirror of focal length 15 cm. Locate the image.

Given: u = −25 cm, f = −15 cm.

$\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = -\frac{1}{15} + \frac{1}{25} = \frac{-5 + 3}{75} = -\frac{2}{75}$

So v = −37.5 cm. The image is real, 37.5 cm in front of the mirror. Magnification m = −v/u = −1.5, so it is inverted and 1.5 times the size.

Example 2: convex mirror

An object is 30 cm in front of a convex mirror of focal length 20 cm. Find the image.

Given: u = −30 cm, f = +20 cm.

$\frac{1}{v} = \frac{1}{20} + \frac{1}{30} = \frac{3 + 2}{60} = \frac{1}{12}$

So v = +12 cm. The image is virtual, 12 cm behind the mirror, upright and diminished.

Example 3: a rear view mirror in metres

A car's convex wing mirror has a radius of curvature of 3.0 m. A truck is 9.0 m behind it. Where is the image?

f = R/2 = +1.5 m and u = −9.0 m.

$\frac{1}{v} = \frac{1}{1.5} + \frac{1}{9.0} = 0.667 + 0.111 = 0.778\ \text{m}^{-1}$

So v ≈ +1.29 m behind the mirror. The image is small and upright, which is why convex mirrors give a wide field of view.

What is the formula of mirror used for: image positions

Concave mirror images by object position
Object position	Image position	Nature
At infinity	At F	Real, inverted, point sized
Beyond C	Between F and C	Real, inverted, smaller
At C	At C	Real, inverted, same size
Between C and F	Beyond C	Real, inverted, larger
At F	At infinity	Real, inverted, very large
Between F and P	Behind the mirror	Virtual, upright, larger

A convex mirror always gives a virtual, upright, smaller image between P and F behind it, wherever the object is. You can check every row of the table with the formula: try u = 2f and you will find v = 2f.

Mistakes and linked formulas

Putting u in as positive. In the New Cartesian convention a real object always has a negative u.
Forgetting the sign of f. A concave mirror of focal length 10 cm has f = −10 cm.
Mixing units. Use cm throughout or m throughout. The calculator on this page expects metres.
Using the lens formula. For lenses the sign between the two terms is a minus; see the formula of lens.

Once you have v, find the size and orientation of the image with the magnification formula of mirror. A lens with one silvered face behaves like a mirror too; the silvering of lens formula shows how. You will find the rest of ray optics through the waves and optics hub.

Frequently asked questions

What is the formula of mirror?

It reads 1/v + 1/u = 1/f. Here u is the object distance, v the image distance and f the focal length, all measured from the pole.

What is the relation between focal length and radius of curvature?

For a spherical mirror with a small aperture, f = R/2. The focus lies halfway between the pole and the centre of curvature.

Why is u negative in the mirror formula?

In the New Cartesian convention, distances measured against the direction of incident light are negative. The object is in front of the mirror, on the side the light comes from, so its distance is negative.

Does the mirror formula work for plane mirrors?

Yes. A plane mirror has an infinite focal length, so 1/f = 0 and v = −u. The image is as far behind the mirror as the object is in front.

Is the derivation of mirror formula the same for a convex mirror?

The steps are the same, using a virtual image behind the mirror. With the sign convention applied, the result is the same equation 1/v + 1/u = 1/f.