Magnification Formula of Mirror: m = −v/u, Signs and Examples

Q: What is the formula of magnification for a mirror?

For a spherical mirror, m = h′/h = −v/u. In terms of focal length, m = f/(f − u) or m = (f − v)/f.

Q: Does magnification have a unit?

Magnification has no unit. It is a ratio of two heights or two distances, so the units cancel.

Q: What does negative magnification mean?

A negative m means the image is inverted relative to the object. For a single mirror, an inverted image is also a real image.

Q: What is the magnification of a plane mirror?

It is +1. The image is the same size as the object, upright and virtual, and the image distance matches the object distance.

Q: Why does the mirror formula for magnification have a minus sign but the lens formula does not?

For a mirror, a real image forms on the same side as the object, so v and u have the same sign. For a lens, a real image forms on the opposite side. The sign conventions then need a minus for mirrors and none for lenses so that m is negative for real, inverted images in both.

The magnification formula of mirror is $m = \frac{h'}{h} = -\frac{v}{u}$ : the height of the image divided by the height of the object, which also equals minus the image distance over the object distance. A value of −2 means the image is twice as tall as the object and upside down. A value of +0.5 means it is half as tall and the right way up.

Magnification is a ratio of two lengths, so it has no unit. The minus sign comes from the New Cartesian rules of the Indian syllabus, which count any height drawn upward from the axis as positive. It makes the sign of m carry the orientation of the image for you, so you never have to guess whether an image is inverted.

Quick facts

Formula
$m = \frac{h'}{h} = -\frac{v}{u}$
With focal length
$m = \frac{f}{f - u} = \frac{f - v}{f}$
Unit
None: a ratio of two lengths
Dimensional formula
[M⁰ L⁰ T⁰]
m negative
Real and inverted image
m positive
Virtual and upright image
Size
$|m| > 1$ enlarged, $|m| = 1$ same size, $|m| < 1$ diminished

The formula of magnification in symbols

Quantities in m = h′/h = −v/u
Symbol	What it is	Sign rule
$m$	Linear magnification	Negative for inverted images
$h'$	Height of the image	Positive above the axis
$h$	Height of the object	Usually positive
$v$	Where the image forms, measured from P	Below zero in front, above zero behind
$u$	Where the object stands, measured from P	Below zero for any real object

Why the minus? A real image from a concave mirror forms in front of the mirror, so v and u are both negative and v/u is positive. Real images from a single mirror are always inverted, so m must come out negative. The minus sign makes that happen. For a virtual image behind the mirror, v is positive and u negative, so −v/u is positive, matching an upright image.

Where the magnification formula of mirror comes from

In a ray diagram, the ray from the top of the object to the pole reflects at an equal angle. That gives two similar triangles, one with sides h and u and one with sides h′ and v. Their ratio gives $\frac{h'}{h} = \frac{v}{u}$ in size. Inserting signs (h′ negative for an inverted image while u and v are both negative) gives the minus sign:

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

Combine this with the mirror formula to remove v. From $\frac{1}{v} = \frac{1}{f} - \frac{1}{u}$ you get $v = \frac{fu}{u - f}$ , and so

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

This form is handy when a question gives the focal length and the object distance and asks only about the image size.

Worked examples on mirror magnification

Example 1: a concave mirror image

A 2.0 cm pin is set up 15 cm from a concave mirror whose focus lies 10 cm from the pole. Find the magnification and the image height.

Given: u = −15 cm, f = −10 cm. First the image distance:

$\frac{1}{v} = -\frac{1}{10} + \frac{1}{15} = -\frac{1}{30} \Rightarrow v = -30\ \text{cm}$

$m = -\frac{v}{u} = -\frac{-30}{-15} = -2$

Image height h′ = m × h = −2 × 2.0 = −4.0 cm. The image is real, inverted and twice as tall. The short cut m = f/(f − u) = −10/5 = −2 agrees.

Example 2: a shop security mirror

A convex mirror in a shop forms an image 8.0 cm behind it of a customer standing 20 cm in front. Find m.

Given: u = −20 cm, v = +8.0 cm.

$m = -\frac{8.0}{-20} = +0.40$

The image is upright and 0.4 times the real size, so the mirror shows a wide area in a small space.

Example 3: a shaving mirror in metres

A concave shaving mirror is curved with R = 0.60 m. A face is held 0.20 m from it. How magnified is the image?

f = −R/2 = −0.30 m and u = −0.20 m.

$m = \frac{f}{f - u} = \frac{-0.30}{-0.30 + 0.20} = \frac{-0.30}{-0.10} = +3.0$

Upright and three times larger, which is why such mirrors are used for close work. The image is virtual and lies 0.60 m behind the mirror.

Reading the sign and size of m

What the value of m tells you
Value of m	Image	Typical mirror
m less than −1	Real, inverted, enlarged	Concave, object between F and C
m = −1	Real, inverted, same size	Concave, object at C
m between −1 and 0	Real, inverted, smaller	Concave, object beyond C
m between 0 and 1	Virtual, upright, smaller	Convex mirror, any object
m = +1	Virtual, upright, same size	Plane mirror
m greater than 1	Virtual, upright, enlarged	Concave, object between P and F

A convex mirror can never produce m greater than 1 or a negative m. If your answer says otherwise, recheck the signs of u and f.

Common errors and connected pages

Dropping the minus sign. For mirrors, m = −v/u. Leaving the sign out flips every orientation.
Using the lens version. For a lens, $m = \frac{v}{u}$ without a minus, as the magnification of lens formula page explains.
Putting heights in with the wrong sign. An inverted image has a negative h′.
Calling an enlarged image real automatically. A concave mirror makes enlarged images that are real or virtual depending on where the object is.

For full image questions, solve the mirror formula first, then apply magnification. The waves and optics hub and the silvering of lens page cover related mirror systems.

Frequently asked questions

What is the formula of magnification for a mirror?

For a spherical mirror, m = h′/h = −v/u. In terms of focal length, m = f/(f − u) or m = (f − v)/f.

Does magnification have a unit?

Magnification has no unit. It is a ratio of two heights or two distances, so the units cancel.

What does negative magnification mean?

A negative m means the image is inverted relative to the object. For a single mirror, an inverted image is also a real image.

What is the magnification of a plane mirror?

It is +1. The image is the same size as the object, upright and virtual, and the image distance matches the object distance.

Why does the mirror formula for magnification have a minus sign but the lens formula does not?

For a mirror, a real image forms on the same side as the object, so v and u have the same sign. For a lens, a real image forms on the opposite side. The sign conventions then need a minus for mirrors and none for lenses so that m is negative for real, inverted images in both.