# Mirror Equation Calculator

Created by Mehjabin Abdurrazaque
Reviewed by Dominik Czernia, PhD and Steven Wooding
Last updated: Nov 01, 2022

Our mirror equation calculator makes it easy to determine the unknown variable among the distances of the object and image from the pole of a mirror, its focal length, and radius of curvature. This tool also helps you find the two types of magnification of mirrors – linear magnification and areal magnification and enables you to avoid the mistakes we make when we apply the Cartesian sign convention for mirrors.

The mirror equation calculator includes the following:

• Concave mirror calculator;
• Convex mirror calculator; and
• Plane mirror calculator.

Do you want to know a lens' focal length? Our focal length calculator discusses the focal length of lenses in detail!

## What is the mirror formula for curved mirrors?

The mirror formula connects distances of the object and image from the pole of a mirror and the mirror's focal length. Here is the equation relating these three variables:

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

where:

• $f$Focal length of the mirror: the distance between the principal focus and the pole of the mirror.
• $v$Image distance: the distance between the image formed and the pole of the mirror.
• $u$Object distance: the distance between the object placed and the pole of the mirror.

The mirror formula is also valid for plane mirrors. A plane mirror's focal length, $f$ is infinity. Thus, its mirror formula becomes:

$\quad \frac{1}{∞}=\frac{1}{v} + \frac{1}{u}$

or,

$\quad v = -u$

✅ The images formed by plane mirrors are always at the same distance from the mirror as the object's distance.

## How to write the mirror equation in terms of the radius of curvature

A mirror's radius of curvature is the radius of the sphere it is a part of. The focal length, $f$, of a mirror is always half of its radius of curvature, $r$:

$\quad f = \frac{r}{2}$

Hence, we can also write the mirror equation as follows:

$\quad \frac{2}{r}=\frac{1}{v} + \frac{1}{u}$

## What are the two types of magnification of a mirror?

The two types of magnification of a mirror are:

• Linear magnification – Ratio of the image's height to the object's height.
• Areal magnification – Ratio of the image's area to the object's area.

Linear magnification, $m_{linear}$, helps us compare the image's size to that of the object in terms of their heights. It is the ratio of the image's height to the object's height.

In terms of $v$ and $u$, we write the linear magnification formula for mirrors as follows:

$\quad m_{linear} = -\frac{v}{u}$

🙋 The linear magnification formula is true for all types of images formed by the convex or concave mirrors – virtual or real, erect or inverted. For real images, linear magnification is negative. It is positive for virtual images.

Areal magnification, $m_{areal}$, tells us how diminished or enlarged the image is compared to the object in terms of its area. It is the ratio of the image's area to the object's area.

We can find the areal magnification using the distances of the object and the image from the pole of a mirror. The areal magnification formula for mirrors in terms of $v$ and $u$ is:

$\quad m_{areal} = \frac{v²}{u²}$

This mirror equation calculator determines both linear and areal magnification for you.

## How to use this concave mirror equation calculator

An object is placed at a distance of 6 cm from a concave mirror with a focal length of 12 cm. Find the position of the image.

To find the position of the image:

1. Choose our concave mirror equation calculator: Select Concave mirror from the drop-down list for Mirror type (by default, the calculator is a concave mirror equation calculator).
2. Enter the object distance $u$ as -6 cm. To make sure you get accurate results, our calculator reminds you to enter a negative value for object distance according to the Cartesian sign convention. Why don't you try it?
3. Enter the focal distance $f$ as -12 cm.

It's that simple! The concave mirror equation calculator displays the image distance $v$, the radius of curvature $r$, areal magnification $m_{areal}$, and linear magnification $m_{linear}$.

## How to use this convex mirror equation calculator

Let's see how to use the calculator for the same problem, but this time for a convex mirror:

1. Change the calculator to a convex mirror equation calculator: Select Convex mirror from the drop-down list for Mirror type.
2. Enter the object distance $u$ as -6 cm.
3. Enter the focal distance $f$ as 12 cm. Note that the focal length is positive here.

The convex mirror equation calculator shows the values for $v$, $r$, $m_{areal}$, and $m_{linear}$.

🙋 Remember, the Cartesian sign convention for mirrors implies that $u$, $f$, and $r$ are negative for a concave mirror. The sign of $v$ for a concave mirror depends if it's a real image (-) or a virtual image (+). For a convex mirror, $v$, $f$, and $r$ are positive, and $u$ is negative.

## How to use plane mirror equation calculator

To use the plane mirror equation calculator:

• Enter the object distance $u$. It should be a negative number.
• The calculator displays $v$, $f$, $r$, $m_{areal}$, and $m_{linear}$.

✅ For a plane mirror, the focal length $f$ and radius of curvature $r$ are infinity; hence, the images formed by plane mirrors are always virtual.

The linear and areal magnifications of the image formed by a plane mirror are 1, i.e., images created by plane mirrors are of the same size as the object.

Now that you've enjoyed using this calculator, you will definitely like our thin lens equation calculator and our lens maker equation calculator. Let's take a look!

## FAQ

### Why is the focal length of a plane mirror infinity?

The reflected rays from a plane mirror neither diverge from a point nor converge onto a point. They are always parallel to one another. For this reason, we consider a plane mirror's focal length as infinity.

We consider a plane mirror as a part of a sphere of infinite radius. It is, therefore, a spherical mirror with an infinite radius of curvature and an infinite focal length.

### What are the positions of images formed by a concave mirror?

The image positions for a concave mirror with the center of curvature at C, the principal focus at F, and the pole at P.

Object position

Image position

at infinity

at F

beyond C

between F and C

at C

at C

between C and F

beyond C

at F

at infinity

between P and F

behind the mirror

### What are the positions of images formed by a convex mirror?

If the object is at infinity, the image forms at the focus behind the mirror. In all other cases, the image forms between the pole P and focus F behind the mirror.

### Why can't a convex mirror form a real image?

The intersection of light rays forms real images. When a concave mirror creates a real image, the reflected rays converge to a point, and we can see an actual image if we place a screen there.

On the other hand, a convex mirror always diverges the incident beam, making the reflected rays diverge from points behind the mirror. They can't converge to produce an actual image outside the mirror. So, the convex mirrors can't form real images - they always form virtual images.

Mehjabin Abdurrazaque
Mirror type
Concave mirror
💡 Remember that we're using Cartesian sign convention. For convave mirrors, the object distance, focal length, and radius of curvature are always negative.
Object distance
in
Image distance
in
Focal length
in
in
Areal magnification
Linear magnification
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