Mirror Equation Calculator

Created by Dominik Czernia, PhD candidate
Last updated: Sep 16, 2022

The mirror equation calculator is a tool designed to help you with the problem of determining focal length or object and image distances for various mirrors. We included three types of mirror formulae:

1. Concave mirror equation
2. Convex mirror equation
3. Plane mirror equation

Read on to find out what a mirror is, learn more about the lens mirror equations, and see some fun facts, such as how mirrors are made and how to tell if a mirror is two-way. Check the thin lens equation calculator to switch the topic to lens instead of mirrors.

What is a mirror?

A mirror definition says that it's a surface that reflects light. We often use mirrors for things like checking one's appearance or for decorative purposes. We also use them in science - we have written more about a specific example in our thin film optical coating calculator.

The three types of mirror surfaces are concave, convex, and planar. Every kind of mirror behaves differently when reflecting light. Play with the mirror equation calculator to answer the question how does a mirror affect the path of light.

How do mirrors work?

Mirrors work by reflecting light off of their surfaces. When light hits a mirror, it bounces off in a predictable way. For the planar mirrors, the reflected angle is the same as the angle that it hit the mirror. We use various units to describe the intensity of light - we explained it in the lux to lumen calculator.

💡 The situation can be more complex for concave and convex mirrors - in such cases, you need to use the spherical mirror equationto predict the reflected light angle correctly.

Concave mirror calculator

A concave mirror is a mirror with a curved inward surface. We sometimes call it a converging mirror because it reflects light to one focal point if an object is beyond the focal point.

We use the following concave mirror equation to calculate the image formation for concave mirrors:

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

where:

• $u$ - object distance (between the object and the pole of the mirror);
• $v$ - image distance (between the image formed and the pole of the mirror); and
• $f$ - focal length of the mirror (distance between the focal point and the pole of the mirror).

For the concave spherical mirror equation, we use the cartesian sign convention, saying that $f$ and $u$ always have positive values, and $v$ might be a positive or negative number.

Convex mirror calculator

As you may have expected, a convex mirror is a mirror with a curved outward surface. It is a diverging mirror with the following convex mirror equation:

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

, so the lens mirror equation is basically the same as for concave mirrors. However, the cartesian sign convention is different in the convex mirror formula. This time, $f$ and $v$ are positive, while $u$ is always negative.

Plane mirror calculator

The plane mirror equation assumes that $f \rightarrow \infty$. The lens mirror equation is then the following:

$u = -v$

The image formed by the plane mirror is on the other side of the mirror but at the same distance from the mirror as the object's distance.

How to use the mirror equation calculator

So, if you now know what the mirror formula for curved mirrors is, let's switch directly to our tool:

1. First of all, select which type of mirror you want to investigate. We prepared the following three sub-calculators:
• Concave mirror calculator;
• Convex mirror calculator; and
• Plane mirror calculator.
2. Input the parameters you know: object or image distances, focal length, or radius of curvature. You can also enter the magnifications (we described the magnification formula for the mirror in the next section).
3. If you provide enough information, our mirror equation calculator will find the missing values.

Magnification formula for a mirror

We use the magnification formula for a mirror to calculate the magnification or image size of an object. It is the following simple expression:

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

, where $m$ is linear magnification. Notice there is a minus sign which results from the cartesian sign notation.

⚠️ Sometimes you might be more interested in the areal magnification, which tells us how diminished or enlarged the image is. The formula is analogous to the linear magnification formula for a mirror but with squared values and without a minus sign.

How are mirrors made?

Mirrors are usually made of metal, glass, or plastic. Only the surface of the mirror is usually coated with a thin layer of silver or another metal, for example, aluminum, to increase its reflectivity.

How to tell if a mirror is two-way

A two-way mirror is a mirror that allows people on both sides to see each other. To test if a mirror is two-way, stand on one side of the mirror and have someone else stand on the other side. If you can see the person on the other side, then the mirror is two-way. If you cannot see the person on the other side, then the mirror is not two-way.

Another common test is to place your fingernail against the reflective surface and check if there is a gap between your nail and the mirror. If there is no gap between your fingernail and its reflection, it might be a two-way mirror. However, it might not work in every case (the first method is definitely more reliable).

Dominik Czernia, PhD candidate
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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